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<title>Functional equation (L-function)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Functional equation (L-function)</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <a href="L-function" title="L-function"><i>L</i>-functions</a> of <a href="Number_theory" title="Number theory">number theory</a> are expected to have several characteristic properties, one of which is that they satisfy certain <b><a href="Functional_equation" title="Functional equation">functional equations</a></b>. There is an elaborate theory of what these equations should be, much of which is still conjectural.

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<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>

<p>A prototypical example, the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> has a functional equation relating its value at the <a href="Complex_number" title="Complex number">complex number</a> <i>s</i> with its value at 1 − <i>s</i>. In every case this relates to some value ζ(<i>s</i>) that is only defined by <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> from the <a href="Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> definition. That is, writing – as is conventional – σ for the real part of <i>s</i>, the functional equation relates the cases

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<p>and also changes a case with

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<p>in the <i>critical strip</i> to another such case, reflected in the line σ = ½. Therefore, use of the functional equation is basic, in order to study the zeta-function in the whole <a href="Complex_plane" title="Complex plane">complex plane</a>.

</p><p>The functional equation in question for the Riemann zeta function takes the simple form

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<annotation encoding="application/x-tex">{\displaystyle Z(s)=Z(1-s)\,}</annotation>

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</math></span><img src="./e239c6565f52a6235a75b6e554f26a1512bce1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.649ex; height:2.843ex;" alt="{\displaystyle Z(s)=Z(1-s)\,}" loading="lazy"></span></dd></dl>

<p>where <i>Z</i>(<i>s</i>) is ζ(<i>s</i>) multiplied by a <i>gamma-factor</i>, involving the <a href="Gamma_function" title="Gamma function">gamma function</a>. This is now read as an 'extra' factor in the <a href="Euler_product" title="Euler product">Euler product</a> for the zeta-function, corresponding to the <a href="Infinite_prime" class="mw-redirect" title="Infinite prime">infinite prime</a>. Just the same shape of functional equation holds for the <a href="Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta function</a> of a <a href="Number_field" class="mw-redirect" title="Number field">number field</a> <i>K</i>, with an appropriate gamma-factor that depends only on the embeddings of <i>K</i> (in algebraic terms, on the <a href="Tensor_product_of_fields" title="Tensor product of fields">tensor product</a> of <i>K</i> with the <a href="Real_number" title="Real number">real field</a>).

</p><p>There is a similar equation for the <a href="Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a>, but this time relating them in pairs:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (s,\chi )=\varepsilon \Lambda (1-s,\chi ^{*})}</annotation>

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</math></span><img src="./6cbaf732bc8cb01afcd74410d0b7c6204511f8bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.243ex; height:2.843ex;" alt="{\displaystyle \Lambda (s,\chi )=\varepsilon \Lambda (1-s,\chi ^{*})}" loading="lazy"></span></dd></dl>

<p>with χ a <a href="Primitive_Dirichlet_character" class="mw-redirect" title="Primitive Dirichlet character">primitive Dirichlet character</a>, χ<sup>*</sup> its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of <a href="Absolute_value" title="Absolute value">absolute value</a> 1, of shape

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<annotation encoding="application/x-tex">{\displaystyle G(\chi ) \over {\left|G(\chi )\right\vert }}</annotation>

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</math></span><img src="./e6d6577e0a28ed45341c5a8fb4dc06ff01a1724a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.221ex; height:6.509ex;" alt="{\displaystyle G(\chi ) \over {\left|G(\chi )\right\vert }}" loading="lazy"></span></dd></dl>

<p>where <i>G</i>(χ) is a <a href="Gauss_sum" title="Gauss sum">Gauss sum</a> formed from χ. This equation has the same function on both sides if and only if χ is a <i>real character</i>, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of <i>Λ</i>(<i>s</i>) at <i>s</i> = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such <i>simple</i> zero can exist (the function is <i>even</i> about the point).

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<div class="mw-heading mw-heading2"><h2 id="Theory_of_functional_equations">Theory of functional equations</h2></div>

<p>A unified theory of such functional equations was given by <a href="Erich_Hecke" title="Erich Hecke">Erich Hecke</a>, and the theory was taken up again in <a href="Tate's_thesis" title="Tate's thesis">Tate's thesis</a> by <a href="John_Tate_(mathematician)" title="John Tate (mathematician)">John Tate</a>. Hecke found generalised characters of number fields, now called <a href="Hecke_character" title="Hecke character">Hecke characters</a>, for which his proof (based on <a href="Theta_function" title="Theta function">theta functions</a>) also worked. These characters and their associated L-functions are now understood to be strictly related to <a href="Complex_multiplication" title="Complex multiplication">complex multiplication</a>, as the Dirichlet characters are to <a href="Cyclotomic_field" title="Cyclotomic field">cyclotomic fields</a>.

</p><p>There are also functional equations for the <a href="Local_zeta-function" class="mw-redirect" title="Local zeta-function">local zeta-functions</a>, arising at a fundamental level for the (analogue of) <a href="Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a> in <a href="%C3%89tale_cohomology" title="Étale cohomology">étale cohomology</a>. The Euler products of the <a href="Hasse%E2%80%93Weil_zeta-function" class="mw-redirect" title="Hasse–Weil zeta-function">Hasse–Weil zeta-function</a> for an <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a> <i>V</i> over a number field <i>K</i>, formed by reducing <i>modulo</i> <a href="Prime_ideal" title="Prime ideal">prime ideals</a> to get local zeta-functions, are conjectured to have a <i>global</i> functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from <a href="Automorphic_representation" class="mw-redirect" title="Automorphic representation">automorphic representation</a> theory seems required to get the functional equation. The <a href="Taniyama%E2%80%93Shimura_conjecture" class="mw-redirect" title="Taniyama–Shimura conjecture">Taniyama–Shimura conjecture</a> was a particular case of this as general theory. By relating the gamma-factor aspect to <a href="Hodge_theory" title="Hodge theory">Hodge theory</a>, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.

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<ul><li><a href="Explicit_formula_(L-function)" class="mw-redirect" title="Explicit formula (L-function)">Explicit formula (L-function)</a></li>

<li><a href="Riemann%E2%80%93Siegel_formula" title="Riemann–Siegel formula">Riemann–Siegel formula</a> (particular approximate functional equation)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/25.15">"§25.15 Dirichlet -functions on NIST"</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Functional_Equation"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FunctionalEquation.html">"Functional Equation"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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