Frobenius distributions of Drinfeld modules over finite fields

We express the weighted class number of Drinfeld $A$-modules of rank two with given characteristic polynomial over the finite field ${\mathbb{F}} _{\mathfrak{p}}=A/{\mathfrak{p}}$ $({\mathfrak{p}} \in\operatorname{Spec}A$, where $A=\mathbb{F} _q[T])$ as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.