TY - JOUR

T1 - Mean square values of Hecke L-series formed with r-th order characters

AU - Diaconu, Adrian

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00222-004-0363-6

DO - 10.1007/s00222-004-0363-6

M3 - Article

AN - SCOPUS:4544236294

VL - 157

SP - 635

EP - 684

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -