Example of Fourier Transform that's not in $L^p$.

Question

For $f \in L^1(\mathbb{R}^d)$, define $\hat{f}(u)=\int f(x)e^{iux}dx$. I know that $\hat{f} \in L^{\infty}$ and that if $f \in L^2$, then $\hat{f} \in L^2$. I also know examples where $\hat{f} \notin L^1$. What is an example of when $\hat{f} \notin L^p$ for some $p > 1$?