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

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 ∈ 1/2 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 Γ₀(N) and prove the analogous result for Jacobi forms.