A product identity for Dirichlet series satisfying Hecke's functional equation

In this paper, we give an analogue of Wilton’s product formula for Dirichlet series that satisfy Hecke’s functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan’s L-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields and Dirichlet L-functions. A $4$-term product identity for Riemann zeta function is also given.