Certain L-functions at s = 1/2

Shin-ichiro Mizumoto

Acta Arithmetica (1999)

  • Volume: 88, Issue: 1, page 51-66
  • ISSN: 0065-1036

Abstract

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Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields.    In this paper we study the central zeros of the following types of L-functions:    (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ),    (ii) the Rankin-Selberg convolution for a pair of Hecke eigenforms for SL₂(ℤ),    (iii) the Dedekind zeta functions.   The paper is organized as follows. In Section 1, the Mellin transform L(s,f) of a holomorphic Hecke eigenform f for SL₂(ℤ) is studied. We note that every L-function in this paper is normalized so that it has a functional equation under the substitution s ↦ 1-s. In Section 2, we study some nonvanishing property of the Rankin-Selberg convolutions at s=1/2. Section 3 contains Kurokawa's result asserting the existence of number fields such that the vanishing order of the Dedekind zeta function at s=1/2 goes to infinity.

How to cite

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Shin-ichiro Mizumoto. "Certain L-functions at s = 1/2." Acta Arithmetica 88.1 (1999): 51-66. <http://eudml.org/doc/207231>.

@article{Shin1999,
abstract = { Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields.    In this paper we study the central zeros of the following types of L-functions:    (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ),    (ii) the Rankin-Selberg convolution for a pair of Hecke eigenforms for SL₂(ℤ),    (iii) the Dedekind zeta functions.   The paper is organized as follows. In Section 1, the Mellin transform L(s,f) of a holomorphic Hecke eigenform f for SL₂(ℤ) is studied. We note that every L-function in this paper is normalized so that it has a functional equation under the substitution s ↦ 1-s. In Section 2, we study some nonvanishing property of the Rankin-Selberg convolutions at s=1/2. Section 3 contains Kurokawa's result asserting the existence of number fields such that the vanishing order of the Dedekind zeta function at s=1/2 goes to infinity. },
author = {Shin-ichiro Mizumoto},
journal = {Acta Arithmetica},
keywords = {central zero; integral representations; nonvanishing; modular -series; Rankin-Selberg convolutions; holomorphic projection operator},
language = {eng},
number = {1},
pages = {51-66},
title = {Certain L-functions at s = 1/2},
url = {http://eudml.org/doc/207231},
volume = {88},
year = {1999},
}

TY - JOUR
AU - Shin-ichiro Mizumoto
TI - Certain L-functions at s = 1/2
JO - Acta Arithmetica
PY - 1999
VL - 88
IS - 1
SP - 51
EP - 66
AB - Introduction. The vanishing orders of L-functions at the centers of their functional equations are interesting objects to study as one sees, for example, from the Birch-Swinnerton-Dyer conjecture on the Hasse-Weil L-functions associated with elliptic curves over number fields.    In this paper we study the central zeros of the following types of L-functions:    (i) the derivatives of the Mellin transforms of Hecke eigenforms for SL₂(ℤ),    (ii) the Rankin-Selberg convolution for a pair of Hecke eigenforms for SL₂(ℤ),    (iii) the Dedekind zeta functions.   The paper is organized as follows. In Section 1, the Mellin transform L(s,f) of a holomorphic Hecke eigenform f for SL₂(ℤ) is studied. We note that every L-function in this paper is normalized so that it has a functional equation under the substitution s ↦ 1-s. In Section 2, we study some nonvanishing property of the Rankin-Selberg convolutions at s=1/2. Section 3 contains Kurokawa's result asserting the existence of number fields such that the vanishing order of the Dedekind zeta function at s=1/2 goes to infinity.
LA - eng
KW - central zero; integral representations; nonvanishing; modular -series; Rankin-Selberg convolutions; holomorphic projection operator
UR - http://eudml.org/doc/207231
ER -

References

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  1. [Br] R. Brauer, A note on zeta-functions of algebraic number fields, Acta Arith. 24 (1973), 325-327. Zbl0272.12011
  2. [De] P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1973), 273-307. 
  3. [Den1] C. Deninger, Motivic L-functions and regularized determinants. I, in: Proc. Sympos. Pure Math. 55, Part I, Amer. Math. Soc., 1994, 707-743. Zbl0816.14010
  4. [Den2] C. Deninger, Motivic L-functions and regularized determinants. II, in: Arithmetic Geometry, F. Catanese (ed.), Symposia Math. 37, Cambridge Univ. Press, 1997, 138-156. Zbl1163.11338
  5. [E-M-O-T1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, 1953. Zbl0051.30303
  6. [E-M-O-T2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, 1954. Zbl0055.36401
  7. [F-H] S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL(2), Ann. of Math. 142 (1995), 385-423. Zbl0847.11026
  8. [Frö] A. Fröhlich, Artin root numbers and normal integral bases for quaternion fields, Invent. Math. 17 (1972), 143-166. Zbl0261.12008
  9. [Ko1] W. Kohnen, Modular forms of half-integral weight on Γ₀(4), Math. Ann. 248 (1980), 249-266. Zbl0416.10023
  10. [Ko2] W. Kohnen, Nonvanishing of Hecke L-functions associated to cusp forms inside the critical strip, J. Number Theory 67 (1997), 182-189. Zbl0896.11019
  11. [Ko-Za] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175-198. Zbl0468.10015
  12. [Ma] H. Maass, Lectures on Modular Functions of One Complex Variable (revised edition), Tata Inst. Fund. Res. Lectures on Math. and Phys. 29, Springer, 1983. 
  13. [Mi] S. Mizumoto, Eisenstein series for Siegel modular groups, Math. Ann. 297 (1993), 581-625; Corrections, Math. Ann. 307 (1997), 169-171. Zbl0786.11024
  14. [Rad] H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169, Springer, 1973. Zbl0253.10002
  15. [Ran] R. A. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions, Proc. Cambridge Philos. Soc. 35 (1939), 351-372. 
  16. [Se] A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47-50. Zbl66.0377.01
  17. [Sha] F. Shahidi, Third symmetric power L-functions for GL(2), Compositio Math. 70 (1989), 245-273. Zbl0684.10026
  18. [Sh] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), 783-804. Zbl0348.10015
  19. [St] J. Sturm, The critical values of zeta functions associated to the symplectic group, Duke Math. J. 48 (1981), 327-350. Zbl0483.10026
  20. [W1] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484. Zbl0431.10015
  21. [W2] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219-307. Zbl0724.11026

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