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: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 [[
The functional equation in question for the Riemann zeta function takes the simple form
:''Z''(''s'') = ''Z''(1 − ''s'')
where ''Z''(''s'') is multiplied by a ''gamma-factor'', involving the [[gamma function]]. This is now read as an 'extra' factor in the [[Euler product]] for the zeta-function, corresponding to the [[infinite prime]]. Just the same shape of functional equation holds for the [[Dedekind zeta function]] of a [[number field]] ''K'', with an appropriate gamma-factor that depends only the embeddings of ''K'' (in algebraic terms, on the [[tensor product of fields|tensor product]] of ''K'' with the [[real field]]).
There is a similar equation for the [[Dirichlet L-function]]s, but this time relating them in pairs:
:<math>\Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^*)</math>
with χ a (primitive) [[Dirichlet character]], χ<sup>*</sup> its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of [[absolute value]] 1, of shape
:<math>G(\chi) \over {\left |G(\chi)\right \vert}</math>
where ''G''(χ) is a [[Gauss sum]] 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).
A unified theory of such functional equations was given by [[Erich Hecke]], and the theory taken up again in ''[[Tate's thesis]]'' by [[John Tate]]. Hecke found generalised characters of number fields. now called [[Hecke character]]s, for which his proof (based on [[theta function]]s) also worked. These characters and their associated L-functions are now understood to be strictly related to [[complex multiplication]], as the Dirichlet characters are to [[cyclotomic field]]s.
There are also functional equations for the [[local zeta-function]]s, arising at a fundamental level for the (analogue of) [[Poincaré duality]] in [[étale cohomology]]. The Euler products of the [[Hasse-Weil zeta-function]] for an [[algebraic variety]] ''V'' over a number field ''K'', formed by reducing ''modulo'' [[prime ideal]]s 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 [[automorphic representation]] theory seems required to get the functional equation. The [[Taniyama-Shimura conjecture]] was a particular case of this as general theory. By relating the gamma-factor aspect to [[Hodge theory]], 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==
*[[explicit formula (L-function)]]
*[[approximate functional equation]]
[[Category:Zeta and L-functions]]
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