Factorization in the extended Selberg class of L ‐functions associated with holomorphic modular forms

Abstract

Let N ∈ ℕ and let χ be a Dirichlet character modulo N. Let f be a modular form with respect to the group Γ0(N), multiplier χ and weight k. Let F be the L ‐function associated with f and normalized in such a way that F (s) satisfies a functional equation where s reflects in 1 – s. The modular forms f for which F belongs to the extended Selberg class S# are characterized. For these forms the factorization of F in primitive elements of S# is enquired. In particular, it is proved that if f is a cusp form and F ∈ S# then F is almost primitive (i.e., that if F = PG is a factorization with P, G ∈ S# and the degree of P is < 2 then P is a Dirichlet polynomial). It is also proved that the conductor of the polynomial factor P is bounded by N. If f belongs to the space generated by newforms and N ≤ 4 then F is actually primitive (i.e., P is a constant) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)