Uniform approximate functional equation for principal L-functions

By Gergely Harcos. 2001

56 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/5093ca8c939da0c58f7235f59b609a5bd70940a8

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In Applied Mathematics

Citations in the corpus

Weak subconvexity for central values of $L$-functions (2008, methodology): “Theorem 1 follows from Theorem 2 by a standard argument using an “approximate functional equation” for L( 1 2 , π) (see for example Harcos [13], Theorem 2.1) and partial summation.”
Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem (2009, methodology): “and we view the space H := � C H (s)ds as a holomorphic fibre bundle with base C.”
On the Voronoĭ formula for GL(n) (2013, background): “The reader may refer to [15] or [16, section 5.”
Hybrid bounds for twisted L-functions (2008, background): “for each pair ð j; kÞ A N2 0, see [ Ha ], Theorem 2.1 and Remark 2.7.”