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Some numerical computations concerning spinor zeta functions in genus $\boldsymbol{2}$ at the central point

Authors: Winfried Kohnen and Michael Kuß
Journal: Math. Comp. 71 (2002), 1597-1607
MSC (2000): Primary 11F46
Published electronically: December 5, 2001
MathSciNet review: 1933046
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Abstract: We numerically compute the central critical values of odd quadratic character twists with respect to some small discriminants $D$ of spinor zeta functions attached to Seigel-Hecke eigenforms $F$ of genus 2 in the first few cases where $F$ does not belong to the Maass space. As a result, in the cases considered we can numerically confirm a conjecture of Böcherer, according to which these central critical values should be proportional to the squares of certain finite sums of Fourier coefficients of $F$.

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  • [1] M. Abramowitz, I. Stegun: Pocketbook of mathematical functions. Verlag Harri Deutsch, (1984). MR 85j:00005b
  • [2] A. Andrianov: Euler products corresponding to Siegel modular forms of genus 2. Russ. Math. Surveys 29, No.3, 45-116 (1974). MR 55:5540
  • [3] S. Böcherer: Bemerkungen über die Dirichletreihen von Koecher und Maass. Math. Gottingensis, Schriftenr. d. Sonderforschungsbereichs Geom. Anal. 68, (1986).
  • [4] S. Böcherer, R. Schulze-Pillot: The Dirichlet series of Koecher and Maass and modular forms of weight 3/2. Math. Z. 209, No.2, 273-287 (1992). MR 93b:11053
  • [5] M. Eichler, D. Zagier: The theory of Jacobi forms. Progress in Mathematics, Vol. 55. Boston-Basel-Stuttgart: Birkhäuser (1985). MR 86j:11043
  • [6] W. Kohnen: On character twists of certain Dirichlet series. Mem. Fac. Sci. Kyushu Univ., vol. 47, 103-117 (1993). MR 94c:11044
  • [7] W. Kohnen, J. Sengupta, A. Krieg: Characteristic twists of a Dirichlet series for Siegel cusp forms. Manuscripta Math. 87, 489-499 (1995). MR 96f:11071
  • [8] W. Kohnen, N.-P. Skoruppa A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math. 95, 541-558 (1989). MR 90b:11050
  • [9] M. Kuß: Die getwistete Spinor Zeta Funktion und die Böcherer Vermutung. Dissertation. (2000)
  • [10] A.F. Lavrik: Functional equations of Dirichlet functions. Soviet Math. Dokl. 7, 1471-1473 (1966). MR 34:4464
  • [11] H. Maass: Ueber eine Spezialschar von Modulformen zweiten Grades. I, II, III Invent. Math. 52, 95-104 (1979), Invent. Math. 53, 249-253, 255-265 (1979). MR 80f:10031; MR 81a:11037; MR 81a:11038
  • [12] N.P. Skoruppa: Computations of Siegel modular forms of genus two. Math. Comput. 58, 381-398 (1992). MR 92e:11041
  • [13] J.L. Waldspurger: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9) 60, 375-484 (1981). MR 83h:10061
  • [14] R. Weissauer: The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

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Additional Information

Winfried Kohnen
Affiliation: Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany

Michael Kuß
Affiliation: Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany

Received by editor(s): October 20, 1999
Received by editor(s) in revised form: January 3, 2001
Published electronically: December 5, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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