  Let $\pi$ be a unitary cuspidal automorphic representation of
$\text{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$. Let $f \geq 1$ be given. We show that
there exists infinitely many primitive even (resp. odd) Dirichlet characters
$\chi$ with conductor co-prime to $f$ such that $L(s, \pi \otimes \chi)$ is
non-vanishing at the central point.
  Our result has applications for the construction of $p$-adic $L$-functions
for $\text{GSp}_{4}$ following Loeffler-Pilloni-Skinner-Zerbes, the Bloch-Kato
conjecture and the Birch-Swinnerton-Dyer conjecture for abelian surfaces
following Loeffler-Zerbes, strong multiplicity one results for paramodular
cuspidal representations of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and the
rationality of the central values of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$
$L$-functions in the remaining non-regular weight case.
