Non-vanishing of $L$-functions of Vector-valued Modular Forms

Abstract

Kohnen proved a non-vanishing result for $L$-functions associated to Hecke eigenforms of integral weights on the full group. In this paper, we show a non-vanishing result for the averages of $L$-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms of weight $k \in \frac{1}{2} \mathbb{Z}$ on the full group. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize the result of Kohnen to $\Gamma_{0}(N)$ and prove the analogous result for Jacobi forms.