Zero set of convolution of integrable function and compactly supported smooth function.

Question

Let $Z_f$ be the zero set of $f\in L^1(\mathbb{R})$ and $S_g$ be the support of $g\in C_c^{\infty}(\mathbb{R}).$ What can we say about the zero set of their convolution $(f*g)$?

$$(f*g)(x)=\int_\mathbb{R}f(x-y)g(y)dy=\int_\mathbb{R}g(x-y)f(y)dy$$