Normal distribution $N(\mu, \sigma^2$) has the moment generating function $$m_X(t) = \exp (\mu t+\frac{\sigma^2t^2}{2})$$ and the characteristic function $$\phi_X(t) = \exp (i \mu t-\frac{\sigma^2t^2}{2})$$ which looks almost the same. In fact, it satisfies the equation $$m_X(it) = \phi_X(t)$$ for all $t\in \mathbb{R}$.
My question : Is there a criterion for a distribution to satisfy $m_X(it) = \phi_X(t)$ ? I'm especially interested in continuous distributions.
I had a course on measure theory, and I'm new to probability theory. I know that moment generating function can be failed to be defined for all $t\in \mathbb{R}$. I saw some examples of moment generating functions and characteristic functions, and all of them satisfy the equation above.