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A new reciprocity formula for Dirichlet $L$-functions associated to anarbitrary primitive Dirichlet character of prime modulus $q$ is established. Wefind an identity relating the fourth moment of individual Dirichlet$L$-functions in the $t$-aspect to the cubic moment of central $L$-values ofHecke-Maass newforms of level at most $q^{2}$ and primitive central character$\psi^{2}$ averaged over all primitive nonquadratic characters $\psi$ modulo$q$. Our formula can be thought of as a reverse version of recent work ofPetrow-Young. Direct corollaries involve a variant of Iwaniec's short intervalfourth moment bound and the twelfth moment bound for Dirichlet $L$-functions,which generalise work of Jutila and Heath-Brown, respectively. This worktraverses an intersection of classical analytic number theory and automorphicforms.

Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications