Functional Equations for Hecke–Maass Series

The Dirichlet (Hecke–Maass) series associated with the eigenfunctions $f$ and $g$ of the invariant differential operator $\Delta_k=-y^2(\partial^2\!/\partial x^2+\partial^2\!/\partial y^2)+ iky\,\partial/\partial x$ of weight $k$ are investigated. It is proved that any relation of the form $(f|_kM)=g$ for the $k$-action of the group $SL_2(\mathbb{R})$ is equivalent to a pair of functional equations relating the Hecke–Maass series for $f$ and $g$ and involving only traditional gamma factors.