How can I show that for random variable $Z$ with Charateristic function $e^{-\vert t \vert+it}$, $E[\vert Z \vert]$ does not exist?

Question

How can I show that for random variable $Z$ with Characteristic function $e^{-\vert t \vert+it}$, $E[\vert Z \vert]$ does not exist? I am not sure how to proceed. Any hints would be highly appreciated.

Answer 1

Suppose for the sake of contradiction that $E(|Z|)<\infty$. It is well-known that in that case, $\phi_Z$ must be differentiable at $0$.

However $\text{Re} (\phi_Z):t\mapsto e^{-|t|}\cos(t)$ is not differentiable at $0$, a contradiction.