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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, the `F33f`_`[L-functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=L-function]`_`f of `F33f`_`[number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_theory]`_`f are expected to have several characteristic properties, one of which is that they satisfy certain `!`F33f`_`[functional equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_equation]`_`f`!. There is an elaborate theory of what these equations should be, much of which is still conjectural.
>>Contents
• `F0af`_`[Introduction`#introduction]`_`f
• `F0af`_`[Theory of functional equations`#theory-of-functional-equations]`_`f
• `F0af`_`[See also`#see-also]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[External links`#external-links]`_`f
-─
>>Introduction
A prototypical example, the `F33f`_`[Riemann zeta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann_zeta_function]`_`f has a functional equation relating its value at the `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f `*s`* with its value at 1 − `*s`*. In every case this relates to some value ζ(`*s`*) that is only defined by `F33f`_`[analytic continuation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Analytic_continuation]`_`f from the `F33f`_`[infinite series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Infinite_series]`_`f definition. That is, writing – as is conventional – σ for the real part of `*s`*, the functional equation relates the cases
σ > 1 and σ < 0,
and also changes a case with
0 < σ < 1
in the `*critical strip`* 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 `F33f`_`[complex plane`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_plane]`_`f.
The functional equation in question for the Riemann zeta function takes the simple form
Z ( s ) = Z ( 1 − − s ) {\\displaystyle Z(s)=Z(1-s)\\,}
where `*Z`*(`*s`*) is ζ(`*s`*) multiplied by a `*gamma-factor`*, involving the `F33f`_`[gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_function]`_`f. This is now read as an 'extra' factor in the `F33f`_`[Euler product`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler_product]`_`f for the zeta-function, corresponding to the `F33f`_`[infinite prime`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Infinite_prime]`_`f. Just the same shape of functional equation holds for the `F33f`_`[Dedekind zeta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dedekind_zeta_function]`_`f of a `F33f`_`[number field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_field]`_`f `*K`*, with an appropriate gamma-factor that depends only on the embeddings of `*K`* (in algebraic terms, on the `F33f`_`[tensor product`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tensor_product_of_fields]`_`f of `*K`* with the `F33f`_`[real field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f).
There is a similar equation for the `F33f`_`[Dirichlet L-functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirichlet_L-function]`_`f, but this time relating them in pairs:`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]
Λ Λ ( s , χ χ ) = ε ε Λ Λ ( 1 − − s , χ χ ∗ ∗ ) {\\displaystyle \\Lambda (s,\\chi )=\\varepsilon \\Lambda (1-s,\\chi ^{*})}
with χ a `F33f`_`[primitive Dirichlet character`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Primitive_Dirichlet_character]`_`f, χ* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of `F33f`_`[absolute value`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Absolute_value]`_`f 1, of shape
G ( χ χ ) | G ( χ χ ) | {\\displaystyle G(\\chi ) \\over {\\left|G(\\chi )\\right\\vert }}
where `*G`*(χ) is a `F33f`_`[Gauss sum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauss_sum]`_`f formed from χ. This equation has the same function on both sides if and only if χ is a `*real character`*, 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 `*Λ`*(`*s`*) at `*s`* = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such `*simple`* zero can exist (the function is `*even`* about the point).
>>Theory of functional equations
A unified theory of such functional equations was given by `F33f`_`[Erich Hecke`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Erich_Hecke]`_`f, and the theory was taken up again in `F33f`_`[Tate's thesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tate's_thesis]`_`f by `F33f`_`[John Tate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=John_Tate_(mathematician)]`_`f. Hecke found generalised characters of number fields, now called `F33f`_`[Hecke characters`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hecke_character]`_`f, for which his proof (based on `F33f`_`[theta functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theta_function]`_`f) also worked. These characters and their associated L-functions are now understood to be strictly related to `F33f`_`[complex multiplication`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_multiplication]`_`f, as the Dirichlet characters are to `F33f`_`[cyclotomic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cyclotomic_field]`_`f.
There are also functional equations for the `F33f`_`[local zeta-functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_zeta-function]`_`f, arising at a fundamental level for the (analogue of) `F33f`_`[Poincaré duality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Poincaré_duality]`_`f in `F33f`_`[étale cohomology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Étale_cohomology]`_`f. The Euler products of the `F33f`_`[Hasse–Weil zeta-function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hasse–Weil_zeta-function]`_`f for an `F33f`_`[algebraic variety`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_variety]`_`f `*V`* over a number field `*K`*, formed by reducing `*modulo`* `F33f`_`[prime ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_ideal]`_`f to get local zeta-functions, are conjectured to have a `*global`* 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 `F33f`_`[automorphic representation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Automorphic_representation]`_`f theory seems required to get the functional equation. The `F33f`_`[Taniyama–Shimura conjecture`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taniyama–Shimura_conjecture]`_`f was a particular case of this as general theory. By relating the gamma-factor aspect to `F33f`_`[Hodge theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hodge_theory]`_`f, 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.
>>See also
• `F33f`_`[Explicit formula (L-function)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Explicit_formula_(L-function)]`_`f
• `F33f`_`[Riemann–Siegel formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann–Siegel_formula]`_`f (particular approximate functional equation)
>>References
`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f "§25.15 Dirichlet -functions on NIST".
>>External links
• `:reference-mathworld-functional-equation`a`:citerefweisstein`a`F33f`_`[Weisstein, Eric W.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_W._Weisstein]`_`f "Functional Equation". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.
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