Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts

Let $Q(n)$ denote the number of partitions of $n$ into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the well-known fact that $Q(n)$ is almost always divisible by $4$. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan's mock theta functions, which in turn gives generating functions for values of certain Dirichlet $L$-functions at nonpositive integers.