•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
•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) = sRt(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
is the functional equation for the standard L-function.
•The “archimedean integral” here is sRe(x)|x|s-1sgn(x)ddx, and
(apparently) the only one that occurs