Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T01:30:11.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluating $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}L$-functions with few known coefficients

Part of: Computational number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2014

David W. Farmer  and
Nathan C. Ryan
David W. Farmer
Affiliation:
American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306, USA email [email protected]
Nathan C. Ryan
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

MSC classification

Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11Y35: Analytic computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 245 - 258
Copyright
© The Author(s) 2014 

References

Andrianov, A. N., ‘Euler products that correspond to Siegel’s modular forms of genus 2’, Uspehi Mat. Nauk 29 (1974) 43–110.Google Scholar
Andrianov, A. N., Quadratic forms and Hecke operators , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 286 (Springer, Berlin, 1987).CrossRefGoogle Scholar
Bloch, S. and Kato, K., ‘ L-functions and Tamagawa numbers of motives’, The Grothendieck Festschrift, Vol. I , Progress in Mathematics 86 (Birkhäuser Boston, Boston, MA, 1990) 333–400.Google Scholar
Böcherer, S., ‘Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe’, J. reine angew. Math. 362 (1985) 146–168.Google Scholar
Böcherer, S., ‘Bemerkungen über die Dirichletreihen von Koecher und Maaß (Remarks on the Dirichlet series of Koecher and Maaß)’, Technical Report, Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 68, 36 S., 1986.Google Scholar
Brian Conrey, J., Farmer, D. W., Keating, J. P., Rubinstein, M. O. and Snaith, N. C., ‘Integral moments of L-functions’, PLMS 91 (2005) 33–104.Google Scholar
Brian Conrey, J., Farmer, D. W. and Zirnbauer, M. R., ‘Autocorrelation of ratios of L-functions’, Commun. Number Theory Phys. 2 (2008) 593–636.Google Scholar
Dokchitser, T., ‘Computing special values of motivic L-functions’, Exp. Math. 13 (2004) 137–149.Google Scholar
Farmer, D. W., Ryan, N. C. and Schmidt, R., ‘Testing the functional equation of a high-degree Euler product’, Pacific J. Math. 253 (2011) 349–366.CrossRefGoogle Scholar
Keating, J. P. and Snaith, N. C., ‘Random matrices and L-functions’, J. Phys. A 36 (2003) 2859–2881; Random matrix theory.CrossRefGoogle Scholar
Klingen, H., Introductory lectures on Siegel modular forms , Cambridge Studies in Advanced Mathematics 20 (Cambridge University Press, Cambridge, 1990).Google Scholar
Kohnen, W. and Kuss, M., ‘Some numerical computations concerning spinor zeta functions in genus 2 at the central point’, Math. Comp. 71 (2002) 1597–1607.Google Scholar
Molin, P., ‘Intégration numérique et calculs de fonctions L’, PhD Thesis, Institut de Mathématiques de Bordeaux, 2010.Google Scholar
Pitale, A., Saha, A. and Schmidt, R., ‘Transfer of Siegel cusp forms of degree 2’, Mem. Amer. Math. Soc., to appear, arXiv:1106.5611.Google Scholar
Rubinstein, M. O., ‘Computational methods and experiments in analytic number theory’, Recent perspectives in random matrix theory and number theory , London Mathematical Society Lecture Note Series 322 (Cambridge University Press, Cambridge, 2005) 425–506.CrossRefGoogle Scholar
Ryan, N. C., ‘Computing the Satake $p$ -parameters of Siegel modular forms’, Preprint, 2004,arXiv:math/0411393.Google Scholar
Ryan, N. C. and Tornaría, G., ‘A Böcherer-type conjecture for paramodular forms’, Int. J. Number Theory 7 (2011) 1395–1411.Google Scholar
Skoruppa, N.-P., ‘Computations of Siegel modular forms of genus two’, Math. Comp. 58 (1992) 381–398.CrossRefGoogle Scholar
Skoruppa, N.-P., ‘Data for rational Siegel eigenforms’, Personal website, 2010.Google Scholar
Wolfram Research Inc. Mathematica (Version 8.0), 2010.Google Scholar