Phase-normalized Fricke functional equation ($\Lambda (f,s)=\Lambda (i^{k}f{|}_k{W}_N,k-s)$)

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.