Meromorphic continuation of Dirichlet series for CM-periods of automorphic functions on GL(2)

I consider expansion at a  CM-point for a Hecke-Maass cusp form. This leads
to a collection of (spherical) coefficients analogous to the classical (unipotent)
Fourier coefficients of automorphic functions on GL(2).  These coefficients were
introduced by H. Petersson, and are connected to special values of L-functions
via the theorem of Waldspurger on the torus period. We prove meromorphic
continuation for a  Dirichlet series build from these coefficients. For the Eisenstein
series, this construction leads to a Double Dirichlet series involving Hecke L-functions
for a (CM) quadratic field.