Mixed moments of the Riemann zeta and Dirichlet L-functions

We prove Motohashi's formula for a mixed second moment of the Riemann zeta function and a Dirichlet L-function attached to a primitive Dirichlet character modulo q is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in \mathbb {N}$$\end{document}. If q is an odd prime, our reciprocity formula is consistent with Motohashi's result in the early 1990s. The cubic moment side features two versions of central L-values of automorphic forms on Gamma 0(q)\H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{0}(q) \backslash \mathbb {H}$$\end{document}. The methods involve a blend of analytic number theory and automorphic forms.