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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 6, pp 1707–1728 | Cite as

On a Hecke-type functional equation with conductor \(\varvec{q=5}\)

  • J. Kaczorowski
  • A. Perelli
Article
  • 29 Downloads

Abstract

We give a complete characterization of the solutions F(s) of the analog in the Selberg class of Hecke’s functional equation of conductor 5, namely
$$\begin{aligned} \left( \frac{\sqrt{5}}{2\pi }\right) ^s \varGamma (s+\mu ) F(s) = \omega \left( \frac{\sqrt{5}}{2\pi }\right) ^{1-s} \varGamma (1-s+\overline{\mu }) \overline{F(1-\overline{s})} \end{aligned}$$
with \(\mathfrak {R}{\mu }\ge 0\) and \(|\omega |=1\). The proof is based on several results from our theory of nonlinear twists of L-functions, applied to obtain a full description of the Euler factor of F(s) at \(p=2\), and then on some ideas from a 1995 paper by J. B. Conrey and D. W. Farmer on converse theorems for Euler products.

Keywords

L-functions Hecke theory Selberg class Converse theorems 

Mathematics Subject Classification

11M41 11F66 

Notes

Acknowledgements

We wish to thank Giuseppe Molteni for suggesting the elegant proof in Section 3 that f(z) is a cusp form. This research was partially supported by the Istituto Nazionale di Alta Matematica, by the MIUR grant PRIN-2015 “Number Theory and Arithmetic Geometry” and by grant 2017/25/B/ST1/00208 “Analytic methods in number theory” from the National Science Centre, Poland.

References

  1. 1.
    Berndt, B.C., Knopp, M.I.: Hecke’s Theory of Modular Forms and Dirichlet Series. World Scientific, Singapore (2008)zbMATHGoogle Scholar
  2. 2.
    Carletti, E., Monti Bragadin, G., Perelli, A.: A note on Hecke’s functional equation and the Selberg class. Funct. Approx. 41, 211–220 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Conrey, J.B., Farmer, D.W.: An extension of Hecke’s converse theorem. Int. Math. Res. Not. 9, 445–463 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hecke, E.: Lectures on Dirichlet Series, Modular Functions and Quadratic Forms. Vanderhoeck & Ruprecht, Göttingen (1983)zbMATHGoogle Scholar
  5. 5.
    Iwaniec, H.: Topics in Classical Automorphic Forms. American Mathematical Society (1997)Google Scholar
  6. 6.
    Kaczorowski, J.: Axiomatic theory of \(L\)-functions: the Selberg class. In: Perelli, A., Viola, C. (eds.) Analytic Number Theory, C.I.M.E. Summer School, Cetraro, Italy, 2002, pp. 133–209. Springer, Berlin (2006)Google Scholar
  7. 7.
    Kaczorowski, J., Molteni, G., Perelli, A.: Linear independence in the Selberg class. C. R. Math. Rep. Acad. Sci. Can. 21, 28–32 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kaczorowski, J., Molteni, G., Perelli, A.: Linear independence of \(L\)-functions. Forum Math. 18, 1–7 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaczorowski, J., Perelli, A.: The Selberg class: a survey. In: Györy, K. et al. (ed.) Number Theory in Progress, Proceedings of the Conference in Honor of A. Schinzel, pp. 953–992. de Gruyter (1999)Google Scholar
  10. 10.
    Kaczorowski, J., Perelli, A.: On the structure of the Selberg class, VI: non-linear twists. Acta Arith. 116, 315–341 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaczorowski, J., Perelli, A.: On the structure of the Selberg class, VII: \(1<d<2\). Ann. Math. 173, 1397–1441 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kaczorowski, J., Perelli, A.: Twists, Euler products and a converse theorem for \(L\)-functions of degree 2. Ann. Sc. Norm. Sup. Pisa (V) 14, 441–480 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kaczorowski, J., Perelli, A.: Twists and resonance of \(L\)-functions, I. J. Eur. Math. Soc. 18, 1349–1389 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaczorowski, J., Perelli, A.: Twists and resonance of \(L\)-functions, II. Int. Math. Res. Notices 7637–7670 (2016)Google Scholar
  15. 15.
    Kaczorowski, J., Perelli, A.: A weak converse theorem for degree \(2\) \(L\)-functions with conductor \(1\). Proc. Res. Inst. Math. Sci. Kyoto 53, 337–347 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ogg, A.: Modular Forms and Dirichlet Series. Benjamin, Amsterdam (1969)zbMATHGoogle Scholar
  17. 17.
    Perelli, A.: A survey of the Selberg class of \(L\)-functions, part I. Milan J. Math. 73, 19–52 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Perelli, A.: A survey of the Selberg class of \(L\)-functions, part II. Riv. Mat. Univ. Parma 3*(7), 83–118 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Perelli, A.: Non-linear twists of \(L\)-functions: a survey. Milan J. Math. 78, 117–134 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Perelli, A.: Converse theorems: from the Riemann zeta function to the Selberg class. Boll. U.M.I. 10, 29–53 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stein, W.: Modular forms, a Computational Approach. American Mathematical Society (2007)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceA. Mickiewicz UniversityPoznanPoland
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenoaItaly

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