Let $X$ be a real random variable defined on a probability space $(\Omega, F, P)$. Define its characteristic function $\phi: \mathbb{R} \to \mathbb{C}$ by $\phi(t) = \mathbb{E}[e^{itX}]$ for every real $t$. Is $\phi$ necessarily analytic on $\mathbb{R}$? In other words, is $\phi$ locally representable by power series in $\mathbb{R}$?
This should be the case if $X$ has moments of all orders. But can this hypothesis be weakened? Specific counterexamples are appreciated.