A correspondence theorem for L-functions and partial differential operators

Given an L-function F(s) from the extended Selberg class, we associate a function $\Phi_F(x,y)$. We show that the functions $\Phi_F(x,y)$ are, in the general case, the analogs of the modular forms associated with the GL(2) L-functions. Indeed, we prove that every $\Phi_F(x,y)$ is eigenfunction of a certain partial differential operator. Moreover, we prove a general correspondence theorem for such L-functions involving the functions $\Phi_F(x,y)$.