Factorization in the extended Selberg class of L -functions associated with holomorphic modular forms
✍ Scribed by G. Molteni
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 189 KB
- Volume
- 282
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
Let N ∈ ℕ and let χ be a Dirichlet character modulo N. Let f be a modular form with respect to the group Γ~0~(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)