Uniform approximate functional equation for principal L-functions
摘要
We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field with unitary central character. We investigate the decay rate of the terms involved using the analytic conductor of Iwaniec and Sarnak as a guideline. Straightforward extensions of the results exist for products of central values. We hope that these formulae will help further understanding of the central values of principal L-functions, such as finding good bounds on their various power means, or establishing subconvexity or nonvanishing results in certain families. A crucial role in the proofs is played by recent progress on the Ramanujan--Selberg conjectures achieved by Luo, Rudnick and Sarnak. The bounds at the non-Archimedean places enter through the work of Molteni.
引用
@article{arxiv.math/0111312,
title = {Uniform approximate functional equation for principal L-functions},
author = {Gergely Harcos},
journal= {arXiv preprint arXiv:math/0111312},
year = {2024}
}
备注
8 pages, LaTeX2e; v2: shorter abstract in paper, recent amsart package used; v3: small alterations in the text (most importantly correcting the definition of the analytic conductor (4)), references updated; to appear soon in Internat. Math. Res. Notices