(Almost) primitivity of Hecke \(L\)-functions (Q2466363)

In 1930's \textit{E. Hecke} [Math. Ann. 112, 664--699 (1936; Zbl 0014.01601), Math. Ann. 114, 1--28 (1937; Zbl 0015.40202), Math. Ann. 114, 316--351 (1937; Zbl 0016.35503) (all three in German) and Lectures on Dirichlet series, modular functions and quadratic forms. Göttingen: Vandenhoeck \& Ruprecht (1983; Zbl 0507.10015)] initiated the study of Dirichlet series satisfying degree 2 functional equation with one gamma factor. In 1980's A. Selberg introduced the Selberg class of \(L\)-functions \(S\), and little later J. Kaczorowski and A. Perelli introduced the extended Selberg class \(S^\sharp\) (see their survey [The Selberg class: a survey. Number theory in progress, Zakopane-Koscielisko, 1997. Vol. 2: Elementary and analytic number theory. Berlin: de Gruyter, 953--992 (1999; Zbl 0929.11028)] for the basic definitions and results concerning \(S\) and \(S^\sharp\)). Still another class of Dirichlet series \(\overline S^\sharp\) was defined in [\textit{J. Kaczorowski}, \textit{G. Molteni}, \textit{A. Perelli}, \textit{J. Steuding} and \textit{J. Wolfart}, Funct. Approximatio, Comment. Math. 35, 183--193 (2006; Zbl 1196.11069)]. The latter class differs from \(S^\sharp\) by the lack of complex conjugation in the functional equation. The same paper contains first results describing mutual relations between Hecke's theory, \(S^\sharp\) and \(\overline S^\sharp\). The present authors go one step further and consider the problem of primitivity and almost primitivity of \(L\)-functions associated to the Hecke cusp forms, i.e. of the Dirichlet series \(L_f(s)= \sum^\infty_{n=1} d(n)n^{-s}\), where \(c(n)\) are such that the function \[ f(\tau)= \sum^\infty_{n=1} c(n)\exp(2\pi in\tau/\lambda), \] defined for \(\tau\in \mathbb C\), \(\text{Im}(\tau)> 0\), satisfies the following functional equations \[ f(\tau+ \lambda)= f(\tau)\quad\text{and}\quad f(-1/\tau)= \varepsilon(i/\tau)^k f(\tau) \] for certain fixed \(k\) and \(\varepsilon\in \{-1,1\}\). They prove (Theorem 1) that \(L_f(s)\) as above is almost primitive (i.e. primitive up to a factor of degree 0) both in \(\overline S^\sharp\) and in \(S^\sharp\) (the last claim only if \(L_f\) belongs to \(S^\sharp\)). A similar, but more involved result concerning the primitivity of \(L_f\) in \(\overline S^\sharp\) and \(S^\sharp\) is proved as well (Theorem 2).