Continuous integrable functions whose Fourier transform are not integrable.

Question

Let $f\in C_0(\mathbb{R})\cap L^1(\mathbb{R})$. Any suggestion for necessary and sufficient condition for $\hat{f}\in L^1(\mathbb{R})$? I mean what proper condition on $f$ makes the following diagram complete?

$$ \hat{f}\in L^1(\mathbb{R}) \Longleftrightarrow (f=?) $$

p.s. $\hat{f}$ is the Fourier transform of $f$.