Mean square values of Hecke L-series formed with r-th order characters

Let L be a number field containing the r-th roots of unity. Starting with the Rankin-Selberg convolution of a metaplectic Eisenstein series on the r-fold cover of GL(2) with itself, we construct a Dirichlet series defined over L whose coefficients involve the r-th order twists of a fixed Hecke L-function. We then observe that a group of functional equations can be naturally associated with this construction. Combining this with the convexity theorem for holomorphic functions of several complex variables, we show that this object, as a function of two complex variables, admits meromorphic continuation to ℂ². As an application, we obtain asymptotic formulae for mean square values of the r-th order twists of an arbitrary Hecke L-function defined over L.