The analytic rank under consideration is the order of 0 at $U=1$ of the $L$-function of the simplest type $L(M, U)$ of an Anderson T-motive $M$ of ordinary rank 1 over $\Bbb F_q(\theta)$. These T-motives are the twisted tensor powers of the Carlitz module. The natural action of $GL_2(\Bbb F_q)$ on the set of the twisted Carlitz modules gives the similar action on $L(M, U)$, this means that we get essentially the same results for the order of 0 of $L(M, U)$ at $U=c$ for any $c\in \Bbb F_q^*$. A...
Source: http://arxiv.org/abs/1205.2900v3