On verifiable functions

The Linnik concept of verifiable functions is investigated in the case of normal distribution $N(\xi,\sigma^2)$. The question of verifiability of some analytic function $f$ is reduced by a method of complexification to that of its $C$-verifiability. A function $f(\xi,\sigma^2)$ is called $C$-verifiable if there exists a non-constant critical function $\varphi$ with power depending on complex $\xi$ and $\sigma^2$ ($\operatorname{Re}\sigma^2$) only through $f$. The paper also contains some necessary conditions for $f$ to be $C$-verifiable or verifiable in the Linnik sense.