Functional equation (L-function): Difference between revisions

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In [[mathematics]], the [[L-function]]s of [[number theory]] have certain '''[[functional equation]]s''', as one of their characteristic properties. There is an elaborate theory of what these should be; much of it still conjectural. For 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 as is conventional σ for the real part of ''s'', the functional equation relates the cases

:&sigma; > 1 and &sigma; < 0,

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in the ''critical strip'' to another such case, reflected in the line &sigma; = &frac12;. 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

:''Z''(''s'') = ''Z''(1 &minus; ''s'')

where ''Z''(''s'') is multiplied by a ''gamma-factor'', involving the [[Gammagamma 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: