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