Uunique Aspect of Characteristic Function of a Standard Normal Distribution

Question

We know that the characteristic function (Fourier Transform) of a standard normal distribution is itself upto a constant (since characteristic function of standard normal $Z$ is $\phi_Z(s) = e^{{-s^2}/{2}}$). I'm curious to know if this is the only distribution having this interesting property. If so, why? I search a bit on the internet. I couldn't find any. If this is already discussed elsewhere, any pointers to those is greatly appreciated.