Editing Functional equation (L-function) (section)

== 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 &minus; ''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,&minus;1}. Then ε must be 1 or &minus;1, and the case of the value &minus;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).