On the orthogonal symmetry of $L$functions of a family of Hecke Grössencharacters
Abstract
The family of symmetric powers of an $L$function associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and padic points of view. Here we examine one explicit such family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by ensembles of orthogonal matrices. We prove an asymptotic formula with power savings for the average of these Lvalues, which reproduces, by a completely different method, an asymptotic formula proven by Greenberg and VillegasZagier. We give an upper bound for the second moment which is conjecturally too large by just one logarithm. We also give an explicit conjecture for the second moment of this family, with power savings. Finally, we compute the one level density for this family with a test function whose Fourier transform has limited support. It is known by the work of Villegas  Zagier that the subset of these Lfunctions from our family which have even functional equations never vanish; we show to what extent this result is reflected by our analytic results.
 Publication:

arXiv eprints
 Pub Date:
 December 2012
 arXiv:
 arXiv:1212.2681
 Bibcode:
 2012arXiv1212.2681C
 Keywords:

 Mathematics  Number Theory;
 11F67 (Primary) 15A52;
 11R42 (Secondary)