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 \(q \in \mathbb {N}\) $q\in \mathbb{N}$ . 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 \(\Gamma _{0}(q) \backslash \mathbb {H}\) ${\mathrm{\Gamma }}_0(q)\backslash \mathbb{H}$ . The methods involve a blend of analytic number theory and automorphic forms.