“…Actually, this is a simple but important special case of a duality principle, first studied by S.Bochner in the 1950's, between Dirichlet series with a general Riemann type functional equation and their Mellin transforms; see, e.g., Kaczorowski-Perelli [24]. Another instance of such a duality is represented by Hecke's correspondence, already mentioned above, between modular forms for the triangle groups G(λ), λ ∈ R, generated by z → z + λ and z → −1/z, and Dirichlet series with a functional equation of type similar to (10); the case λ = 1 corresponds to the modular forms for Γ. The more interesting part of such a correspondence is that Hecke managed to compute exactly the dimension of the resulting spaces W , again arguing on the modular forms side.…”