Phase-normalized Fricke functional equation
= Phase-normalized Fricke functional equation
{title2=$\Lambda(f,s)=\Lambda(i^kf|_kW_N,k-s)$}
With the <determinant-normalized slash operator>, the map $Jf=i^kf|_kW_N$ 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.