We investigate the relations for L-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and halfintegral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms defines an automorphic distribution for the group ∆(N ) = ñ(1),ñ(N ) , and hence its Poisson transform, which is given by F α (z) in (III), is a Maass form automorphic for ∆(N ). The transformation formula (0.3) is derived from the comparison of the Poisson transforms of both sides of the summation formula.The group ∆(N ) is a subgroup of the lift Γ 0 (N ) of Γ 0 (N ) to G. In §4 we prove a converse theorem that gives a condition for the automorphic distribution for ∆(N ) associated with the Ferrar-Suzuki summation formula to be automorphic for the larger group Γ 0 (N ) in terms of twists of the corresponding L-functions by Dirichlet characters (Theorem 4.2). By the Poisson transformation, this yields immediately a converse theorem for Maass forms for Γ 0 (N ) (Theorem 4.3), which is an analogue of the converse theorem of Weil [49] for holomorphic modular forms of integral weight (in the case of even ℓ) and its generalization by Shimura [40, Section 5] to holomorphic modular forms of half-integral weight (in the case of odd ℓ). A merit of our approach is to keep calculations involving the Whittaker function to a minimum. The information on the Whittaker function we need is only the fact that the Whittaker function appears as the Fourier transform of the Poisson kernel (see (2.14)). We say now a few words about the assumptions in the converse theorem. In our converse theorem some analytic properties (functional equations, location of poles and so on) are assumed for the L-functions twisted by Dirichlet characters of prime modulus including the principal characters. In the original converse theorem of Weil for holomorphic modular forms ([49]), the assumptions are imposed for the L-functions twisted only by primitive Dirichlet characters of prime modulus. As pointed out in Gelbart-Miller [9, §3.4], it has been considered difficult to transfer the argument of Weil to the Maass form case. In [2] and [3], Diamantis and Goldfeld proved a converse theorem for double Dirichlet series by considering the twists of Dirichlet series by Dirichlet characters including imprimitive characters (the principal characters), namely by the method originally found by Razar [30] for holomorphic modular forms. We follow the approach of Razar and Diamantis-Goldfeld. In a paper [25] that appeared very recently, Nuerurer and Oliver succeeded in modifying the argument of Weil to get a c...