Abstract.
Let f be a cusp form of the Hecke space \({\frak M}_0(\lambda,k,\epsilon)\) and let L f be the normalized L-function associated to f. Recently it has been proved that L f belongs to an axiomatically defined class of functions \(\bar{\cal S}^\sharp\). We prove that when λ ≤ 2, L f is always almost primitive, i.e., that if L f is written as product of functions in \(\bar{\cal S}^\sharp\), then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if \(\lambda\notin\{\sqrt{2},\sqrt{3},2\}\) then L f is also primitive, i.e., that if L f = F 1 F 2 then F 1 (or F 2) is constant; for \(\lambda\in\{\sqrt{2},\sqrt{3},2\}\) the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L f belongs to the more familiar extended Selberg class \({\cal S}^\sharp\) is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in \({\cal S}^\sharp\).
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Molteni, G., Steuding, J. (Almost) primitivity of Hecke L-functions. Mh Math 152, 63–71 (2007). https://doi.org/10.1007/s00605-007-0454-8
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DOI: https://doi.org/10.1007/s00605-007-0454-8