Zeros of the Zak transform on locally compact Abelian groups

Let $G$ be a locally compact abelian group. The notion of Zak transform on $L^2(\mathbb{R}^d)$ extends to $L^2(G)$. Suppose that $G$ is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for $L^2(\mathbb{R})$, we then prove that if $f \in L^2(G)$ is such that its Zak transform $Z f$ is continuous on $G \times \widehat{G}$, then $Z f$ has a zero.