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.

For a weight-$k$ cusp form on ${\Gamma }_1(N)$, use the determinant-normalized slash operator and set $Jf=i^{k}f{|}_k{W}_N$, with $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$. Then $J^2=1$. DefineFor $F(y)=f(iy/\sqrt{N})$ and $G(y)=(Jf)(iy/\sqrt{N})$, direct substitution in the slash operator for modular forms gives $F(y)=y^{-k}G(1/y)$. The Mellin transform and the change of variable $y\mapsto 1/y$ therefore giveBoth integrals are entire in $s$, by exponential cusp form decay at both endpoints after inversion. If $Jf=\epsilon f$, then $\epsilon \in \{1,-1\}$ and ${\Lambda }_N(f,s)=\epsilon {\Lambda }_N(f,k-s)$. The factor $i^k$ is part of this phase convention; changing it changes the sign in the functional equation.