Automorphic pairs of distributions on $\mathbb{R}$, and Maass forms of real weights

We give a correspondence between automorphic pairs of distributions on $\mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $\mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,\mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weights.