Let $X$ be a real random variable with characteristic function $\phi$. Then \begin{align} \phi(2\pi) = Ee^{i2\pi X} = E((e^{i\pi})^2)^X = E((-1)^2)^X = 1. \end{align} There have to be something wrong with the procedure because that equality is not true when, for example, $X\sim N(0,1)$ or $X\sim U(-1,1)$. (In those laste cases $\phi(t)=e^{-t^2/2}$ and $\phi(t)=\sin(t)/t$.)
Can someone help me clear my mistake?