Expected Values of $L$-functions Away from the Central Point

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindel\"of hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.