The functional equation of the twisted spinor zeta-function and Böcherer's conjecture
\textit{A. N. Andrianov} [Russ. Math. Surv. 29, No. 3, 45--116 (1974); translation from Usp. Mat. Nauk 29, 43--110 (1974; Zbl 0304.10020)] showed that the spinor zeta function \(Z_F(s)\) of a Siegel modular form \(F\) of genus 2 and weight \(k\) possesses a meromorphic continuation and satisfies a functional equation under \(s\mapsto 2k-2-s\). Later, \textit{W. Kohnen}, \textit{A. Krieg} and \textit{A. Sengupta} [Manuscr. Math. 87, 489--499 (1995; Zbl 0879.11021)] generalized this to twists \(Z_F(s,\chi)\), whenever \(\chi\) is a Dirichlet character such that \(\chi^2\) is primitive and the first Fourier-Jacobi coefficient \(f_1\neq 0\). In his thesis the author drives the analogous result under the weaker assumption that \(\chi\) is primitive. NEWLINENEWLINENEWLINEMoreover, the author investigates the Böcherer conjecture saying that the central critical value \(Z_F(k-1,\chi_D)\), \(\chi_D= (\frac{D}{\cdot})\), is basically a square. This conjecture is numerically verified in several cases.
scientific article; zbMATH DE number 1620619