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Non-Vanishing of Quadratic Twists of Modular L-Functions

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Non-vanishing of L-Functions and Applications

Part of the book series: Progress in Mathematics ((PM,volume 157))

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Abstract

Let f be a holomorphic cusp form for Γ0(N) of weight 2 and character ∈. We assume that f is a normalized newform for the Hecke operators. Denote by L(s, f) the L-function attached to f. For Re(s) > 3/2, it is given by an absolutely convergent Dirichlet series L\left( {s,f} \right) = \sum\limits_{{n = 1}}^{\infty } {\frac{{a\left( n \right)}}{{{n^{s}}}}} .

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References

  1. P. T. Bateman and S. Chowla, Averages of character sums, Proc. Amer. Math. Soc., 1 (1950), 781–787.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Friedberg and J. Hoffstein, Non-vanishing theorems for automorphic L-functions on GL (2), Annals of Math., 142 (1995), 385–423.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. S. Fainleib and O. Saparnijazov, Dispersion of real character sums and the moments of L(l, x), (Russian) Izv. Akad. Nauk. USSR, Ser. Fiz.-Mat. Nauk, 19 (1975), 24–29.

    MathSciNet  MATH  Google Scholar 

  4. H. Iwaniec, On the order of vanishing of modular L-functions at the critical point, Sém. de Théorie des Nombres Bordeaux, 2 (1990), 365–376.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Jutila, On character sums and class numbers, J. Number Theory, 5 (1973), 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  6. H.L. Montgomery and R.C. Vaughan, Mean values of character sums, Canadian J. Math., 31 (1979), 476–487.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Kumar Murty, A non-vanishing theorem for quadratic twists of modular L-functions, preprint, 1991.

    Google Scholar 

  8. M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular L-series, Annals of Math., 133 (1991), 447–475.

    Article  MATH  Google Scholar 

  9. V. Kumar Murty and T. Stefanicki, Non-vanishing of quadratic twists of L-functions attached to automorphic representations of GL(2) over Q, preprint, 1994.

    Google Scholar 

  10. R. Rankin, Sums of powers of cusp form coefficients II, Math. Ann., 272 (1985), 593–600.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Rohrlich, Non-vanishing of L-functions for GL 2, Invent. Math., 97 (1989), 381–403.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Shimura, On the periods of modular forms, Math. Ann., 229 (1977), 211–221.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Waldspurger, Sur les valeurs de certaines fonctions L automorphe en leur centre de symétrie, Comp. Math., 54 (1985), 173–242.

    MathSciNet  MATH  Google Scholar 

  14. J. Waldspurger, Correspondances de Shimura et quaternions, Forum Math., 3 (1991), 219–307.

    Article  MathSciNet  MATH  Google Scholar 

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© 1997 Springer Basel AG

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Murty, M.R., Murty, V.K. (1997). Non-Vanishing of Quadratic Twists of Modular L-Functions. In: Non-vanishing of L-Functions and Applications. Progress in Mathematics, vol 157. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8956-8_7

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  • DOI: https://doi.org/10.1007/978-3-0348-8956-8_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-5801-3

  • Online ISBN: 978-3-0348-8956-8

  • eBook Packages: Springer Book Archive

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