The Selberg $Z$-function and the Lindelöf conjecture

It is proved that, under some assumptions, the Selberg $Z$-function $Z(s)$ is of order $t^\varepsilon/(\sigma-\frac12)$ in a sufficiently small neighborhood of the critical straight line $\sigma>\frac12$, $t\ge1$, and $\varepsilon>0$ is an arbitrary small but fixed.