  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.
