Mellin transform of a cusp-form L-function

For $f(\tau )={\sum }_{n\ge 1}{a}_n{q}^n$, termwise integration initially givesA Fricke transformation converts the behavior near zero into cusp decay at infinity, giving analytic continuation and a functional equation for the completed L-function.

With the determinant-normalized slash operator, the map $Jf=i^{k}f{|}_k{W}_N$ satisfies $J^2=1$. The scaled imaginary-axis functions satisfy $F(y)=y^{-k}G(1/y)$, and their Mellin transforms give the displayed entire functional equation. The phase must be specified; omitting it changes the sign in odd or twice-odd weights.