## Weighted averages of modular $L$-values

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- by Andrew Knightly and Charles Li PDF
- Trans. Amer. Math. Soc.
**362**(2010), 1423-1443 Request permission

## Abstract:

Using an explicit relative trace formula on $\operatorname {GL}(2)$, we derive a formula for averages of modular $L$-values in the critical strip, weighting by Fourier coefficients, Hecke eigenvalues, and Petersson norms. As an application we show that a GRH holds for these averages as the weight or the level goes to $\infty$. We also use the formula to give explicit zero-free regions of the form $|\operatorname {Im}(s)|\le \tau _0$ for some particular modular $L$-functions.## References

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## Additional Information

**Andrew Knightly**- Affiliation: Department of Mathematics and Statistics, Neville Hall, University of Maine, Orono, Maine 04469-5752
**Charles Li**- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- Received by editor(s): January 31, 2008
- Published electronically: September 25, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**362**(2010), 1423-1443 - MSC (2000): Primary 11F11, 11F30, 11F67, 11F70
- DOI: https://doi.org/10.1090/S0002-9947-09-04923-X
- MathSciNet review: 2563735