Let α>0 and let g∈L₁(R) be a continuous function, whose Fourier transform is ĝ(ω)=Ce^−γω2 e^−2πiδω∏ν=1∞ [Formula presented] where C>0 γ⩾0 δ,δ_ν,λ_j∈R ∑_ν=1 ^∞δ_ν ²<∞ m∈Z₊. We prove that its Zak transform Z_αg(x,ω)=∑_k∈Zg(x+αk)e^−2πikαω has only one zero (x^∗, Formula presented] in the fundamental domain [0,α)×0, [Formula presented]. In particular, the result is valid for totally positive functions. Earlier it was known for such functions without the factor e^−γω2 . We also establish simplicity of the zero with respect to each variable and give the applications to Gabor analysis. The described class of functions is closed under convolution.