Analytic
Continuation of L-functions

•__GL(2) example__: one has (say, for GL(2,Z) automorphic forms)

•Formally, we would like to integrate t(x) against the measure |x|s-1dx. However,
there are potential
singularities at x = 0 and 1. A priori,
distributions can only be integrated against smooth functions of compact support.

•If t(x) is
cuspidal then c0 = 0 and the Fourier
series oscillates a lot near x = 1. More importantly, t(x) has bounded antiderivatives of arbitrarily high
order. This allows one to make sense of the integral when Re s is large or
small.

•Since x = 1 and x = 0 are related by x a 1/x, the same is
true near zero.

•Thus the Mellin transform Mt(s) = sR t(x)|x|s-1dx is
holomorphically defined as a pairing of distributions. It
satisfies the identity Mt(s) = Mt(1-s+2n).

•One computes straightforwardly, term by term, that

which
is the functional equation for the standard L-function.

•The “archimedean integral” here is sR e(x)|x|s-1 sgn(x)d dx, and
(apparently) the only one that occurs
in general.