Functional equation (L-function): Difference between revisions

Revision as of 06:03, 20 September 2010

Revision as of 01:54, 10 November 2010

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{{Unreferenced|date=December 2009}}

In [[mathematics]], the [[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. 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{{Ndash}}as is conventional{{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

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:<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:

:<math>\Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^*)</math>

with χ a [[primitive Dirichlet character]], χ^* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of [[absolute value]] 1, of shape