Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

Abstract

We prove a nonvanishing result for Koecher-Maass series attached to Siegel cusp forms of weight k and degree n in certain strips on the complex plane. When n = 2, we prove such a result for forms orthogonal to the space of the SaitoKurokawa lifts 'up to finitely many exceptions', in bounded regions.

Koecher–Maass series have infinitely many critical zeros

“…So, for a fixed 𝑚 ⩾ 1 and 𝛼 very close to 𝜋∕2, we have < 1 2 𝐹(𝑇), we arrive at a contradiction and it completes the proof of the theorem.…”

“…Koecher–Maass series attached to Siegel modular forms and its properties have been well studied [6]. This series has been studied to establish several other results, such as converse theorem for Siegel modular forms, non‐vanishing of Koecher–Maass series inside a strip [1] and many others. It is not known that Koecher–Maass series are genuine $$L$$‐functions nor if they are expected to satisfy the Riemann hypothesis.…”

Simultaneous nonvanishing of products of L-functions associated to elliptic cusp forms

“…However, Kohnen ( [7]) showed that for k sufficiently large there exists a Hecke eigenform f of weight k such that L * (f, s) = 0 at any point on the line segments Im(s) = t 0 , k−1 2 < Re(s) < k 2 − ǫ, k 2 + ǫ < Re(s) < k+1 2 , for any given real number t 0 and a positive real number ǫ. This result and its method inspired various works on non-vanishing of L-values for different kinds of modular forms (see [2,5,8]). This paper concerns the non-vanishing of the product L * (f, s)L * (f, w) (s, w ∈ C) on average.…”

On Hecke $L$-functions attached to half-integral weight modular forms

“…2010 Mathematics Subject Classification Primary 11F37, Secondary 11F66 Finally, addressing the reader interested in Siegel modular forms, we would like to mention the papers [3,5] where corresponding non-vanishing results for Koecher-Maass series are given.…”