(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)

Let $f:\mathbb{R}\to [0,\infty)$ be such that (a) $\int_{\mathbb{R}} f(x) dx = 1$, (b) $|\widehat{f}(t)|=0$ for all real $t$ with $|t|>1$. What is the choice of $f$ such that $$\int_{\mathbb{R}} |x| f(x) dx$$is minimal? What is that minimum?

Remarks:

1. It is easy to see that we can assume $f$ to be an even function.
2. Yes, this seems to be yet another incarnation of the uncertainty principle.