Rankin-Selberg method for real analytic cusp forms of arbitrary real weight
Let \(\Xi\) be a unitary \(m\)-dimensional multiplier system of real weight \(r\) for \(\Gamma=SL(2,\mathbb{Z})\) and \(F(z)\) be a real-analytic cusp form with respect to \(\Xi\) having values in \(\mathbb{C}^ m\) and which is normalized, i.e. \(\int_{D_ \Gamma}| F(z)|^ 2\;d\mu(z)=1\), \(d\mu\) is the hyperbolic measure, \(D_ \Gamma\) a fundamental domain. We establish a meromorphic continuation for the Rankin-Selberg zeta functions \[ R_{\pm,F}(s):=\sum^ m_{j=1}\quad\sum_{n\equiv\alpha_ j\bmod 1, \pm n>0}|\rho_{j,F}(n)|^ 2/| n|^{s-1} \] where \(\rho_{j,F}(n)\) are the Fourier coefficients of \(F\) and \(\alpha_ j\) are the cusp parameters. This is done via a modification of the Rankin- Selberg method. We obtain a functional equation of the form \(\mathbb{R}_ F(s)=\mathbb{C}_ F(s)\mathbb{R}_ F(1-s)\), where \(\mathbb{C}_ F(x)\) is a 2-by-2 matrix and \(\mathbb{R}_ F(s)\) is the vector \((R_{+,F}(s),R_{-,F}(s))^ T\). \(\zeta(2s)R_{\pm,F}(s)\) are holomorphic with the exception of a simple pole at \(s=1\). The components of \(\mathbb{C}_ F(s)\) contain Mellin transforms of the square of some Whittaker functions. The analytic properties of these are studied to some extent.
scientific article; zbMATH DE number 4216102
scientific article