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In [[mathematics]], the [[L-function|''L''-function]]s of [[number theory]] are expected to have several characteristic properties, one of which is that they satisfy certain '''[[functional equation]]s'''. There is an elaborate theory of what these equations should be, much of which is still conjectural.

== Introduction ==

A prototypical example, the [[Riemann zeta function]] has a functional equation relating its value at the [[complex number]] ''s'' with its value at 1 − ''s''. In every case this relates to some value ζ(''s'') that is only defined by [[analytic continuation]] from the [[infinite series]] definition. That is, writing{{spaced ndash}}as is conventional{{spaced ndash}}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 [[complex plane]].

The functional equation in question for the Riemann zeta function takes the simple form

:<math>Z(s) = Z(1-s) \, </math>

where ''Z''(''s'') is ζ(''s'') 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 on the embeddings of ''K'' (in algebraic terms, on the [[tensor product of fields|tensor product]] of ''K'' with the [[real number|real field]]).

There is a similar equation for the [[Dirichlet L-function]]s, but this time relating them in pairs:<ref>{{cite web|url=https://dlmf.nist.gov/25.15 |title=§25.15 Dirichlet -functions on NIST}}</ref>

:<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).

== Theory of functional equations ==

A unified theory of such functional equations was given by [[Erich Hecke]], and the theory was taken up again in [[Tate's thesis]] by [[John Tate (mathematician)|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)]]
*[[Riemann–Siegel formula]] (particular approximate functional equation)

==References==
{{Reflist}}

== External links ==
*{{MathWorld|FunctionalEquation|Functional Equation}}

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{{DEFAULTSORT:Functional Equation (L-Function)}}
[[Category:Zeta and L-functions]]
[[Category:Functional equations]]

Functional equation (L-function) - Revision history

imported>GregariousMadness at 23:22, 28 December 2024