Characteristic function vs Moment generating function

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 $tin mathbb{R}$.

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 $tin mathbb{R}$. I saw some examples of moment generating functions and characteristic functions, and all of them satisfy the equation above.