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



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.


L-functions Hecke theory Selberg class Converse theorems 

Mathematics Subject Classification

11M41 11F66 



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.


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© 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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