Uniform approximate functional equation for principal L-functions

Abstract

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.

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

Cited by in corpus (25)

• Nodal domains of Maass forms I
• Twisted L-functions over number fields and Hilbert's eleventh problem
• An additive problem in the Fourier coefficients of cusp forms
• Spectre automorphe des variétés hyperboliques et applications topologiques
• Subconvexity for modular form L-functions in the t aspect
• A non-split sum of coefficients of modular forms
• Applications of analytic newvectors for $\mathrm{GL}(n)$
• New bounds for automorphic L-functions
• Subconvexity for a double Dirichlet series
• A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions
• Nodal domains of Maass forms II
• Shifted convolution sums and Burgess type subconvexity over number fields
• Weak subconvexity for central values of $L$-functions
• Beyond the spherical sup-norm problem
• A new zero-free region for Rankin-Selberg $L$-functions
• Weighted fourth moments of Hecke zeta functions with groessencharacters
• The number of coefficients of automorphic $L$-functions for $G{L}_m$ of same signs
• Explicit Upper Bounds for L-functions on the critical line
• A note on the number of coefficients of automorphic $L-$functions for $G{L}_m$ with same signs
• On Fourier coefficients of modular forms of half integral weight at squarefree integers
• Subconvexity of a double Dirichlet series over the Gaussian field
• A Bessel delta-method and exponential sums for GL(2)
• Integral presentations of the shifted convolution problem and subconvexity estimates for ${\mathrm{GL}}_n$-automorphic $L$-functions
• First moment of Rankin-Selberg central L-values and subconvexity in the level aspect
• The Selberg trace formula as a Dirichlet series