Cubic twists of GL(2) automorphic L-functions

Abstract

Let \(K=\mathbb{Q}(\sqrt{-3})\) and let π be a cuspidal automorphic representation of \(GL(2,\mathbb{A}_K)\). Consider the family of twisted L-functions L(s,π⊗χ) where χ ranges over the cubic Hecke characters of K. In this paper the mean value of this family of L-functions is computed; the result is consistent with the generalized Lindelof hypothesis. From this mean value result a nonvanishing theorem is established: for given s there are infinitely many cubic twists such that the L-value at s is nonzero. At the center of the critical strip the number of such characters of norm less than X is \(\gg X^{1/2-\epsilon}\). These results are obtained by introducing and studying three different families of weighted double Dirichlet series. These series are related by functional equations, some of which are obtained through the study of higher metaplectic Eisenstein series and the Hasse-Davenport relation. The authors establish the continuation of such series and then obtain their main result by Tauberian methods.