Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $s$ be a complex variable. Consider the following integral $$F(s)=\int_{\mathbb{R}^*}\chi(x)|x|^{ns}\phi(x)\frac{dx}{x}$$

This is a Tate-type integral at real place. $F(s)$ converges in some right half complex plane and has a meromorphic continuation to the whole plane. The poles of $F(s)$ are responsible for the poles of the $L$ factor of some character.

But here I'm concerning the zeros of $F(s)$ at some right half plane. My question is: may I find some Schwartz function $\phi$ so that $F(s)$ is not identically zero, but has infinitely many zeros of the form $s_0, s_0+2/n, s_0+4/n,....$?

Thank you.