The eighth moment of Dirichlet L-functions

We prove an asymptotic for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q  , over all moduli q⩽Qq⩽Q and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2)GL(2)L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24 024 appears as a factor in the leading order term of the eighth moment.