A note on Hecke's functional equation and the Selberg class
A classical result by \textit{E. Hecke} [Lectures on Dirichlet series, modular functions and quadratic forms. Göttingen: Vandenhoeck \& Ruprecht (1983; Zbl 0507.10015)] states that the \(\mathbb C\)-space of solutions of the functional equation \[ \Biggl({\lambda\over 2\pi}\Biggr)^s \Gamma(s)\phi(s)= \omega\Biggl({\lambda\over 2\pi}\Biggr)^{k-s} \Gamma(k-s) \phi(k-s)\tag{1} \] (\(\lambda\), \(k>0\), \(\omega=\pm 1\)) in Dirichlet series \(\phi(s)\) with a finite abscissa of convergence and a finite order meromorphic continuation to \(\mathbb C\) with at most a simple pole at \(s=k\) has a finite dimension if and only if \(0<\lambda< 2\). Moreover, Hecke gave an explicit formula for the dimension in case when it is finite. After normalizing \(\phi(s)\) by writing \(H(s)=\phi(s (k-1)/2)\) one obtains a solution of a functional equation of type (1) with one (shifted) \(\Gamma\)-factor relating \(H(s)\) to \(H(1-s)\). This is very similar to the situation of (a special case of) degree 2 functional equation in the extended Selberg class \(S^\sharp\) (cf. [\textit{J. Kaczorowski} and \textit{A. Perelli}, The Selberg class: a survey. Number theory in progress, Vol. 2 (Zakopane-Koscielisko, 1997), 953--992 (1999; Zbl 0929.11028)] for the basic definitions and results concerning \(S^\sharp\)). The only differences lie in the location of a possible pole and the lack of complex conjugation in the Hecke theory. Let \(S^\sharp(Q, \mu,\omega)\), where \(Q>0\), \(\text{Re}R(\mu)\geq 0\), \(|\omega|= 1\) denote the space of solutions \(\phi\in S^\sharp\) of the equation \[ \Phi(s)= \omega\overline{\Phi(1-\overline s)} \] where \[ \Phi(s)= Q^s\Gamma(s+ \mu)\phi(s). \] The main result of the present paper is a counterpart of Hecke's theorem and states that \(\dim_{\mathbb R}S^\sharp(Q,\mu,\omega)\) is finite if and only if the conductor \(q:= (2\pi Q)^2\) does not exceed 4. If \(q>4\), the space \(S^\sharp(Q,\mu,\omega)\) contains uncountably many linearly independent functions.
Dirichlet series with functional equation
scientific article; zbMATH DE number 5654319