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We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functionstwisted by primitive even Dirichlet characters of prime-power conductor, atarbitrary points within the critical strip, by derivation of special Voronoisummation formulae. Our calculation is novel in that the twisted sum can beexpressed in terms of the average itself, and also that it sees the derivationof various new summation formulae in the setting of prime-power modulus. Oneconsequence, as we explain, is to show the analytic continuation and additivesummation formulae for hyper-Kloosterman Dirichlet series associated to$\operatorname{GL}_n$-automorphic $L$-functions.

Dirichlet twists of $\operatorname{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series