The \(GL(3)\) Mellin transform for twisted non-cuspidal forms of higher level
This paper is devoted to the study of Dirichlet series whose coefficients are themselves essentially of the form \(\mu\mapsto L(s,\chi_{\mu^2})\) where \(\chi_{\mu^2}\) is the cubic residue character \(\text{mod } \mu^2\). The authors demonstrate that the Dirichlet series with twists can be represented as Mellin transforms of a residue of an Eisenstein series on the three-fold metaplectic cover of \(GL(3)\). They can prove both the analytic continuation and a `functional equation'; the latter involve, at least in a special case, a Dirichlet series where classical Kloosterman sums appear. They can also deduce asymptotic estimates for the average size of the \(L\)-functions. They quote this but defer the proof to a joint paper with \textit{J. Hoffstein}.
scientific article; zbMATH DE number 1270359