  We investigate the relations for $L$-functions satisfying certain functional
equation, summationa formulas of Voronoi-Ferrar type and Maass forms of
integral and half-integral 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 studied
by Peter and the fourth author.
