  We introduce and study a class of multivariate rational functions associated
with hyperplane arrangements, called flag Hilbert-Poincar\'e series. These
series are intimately connected with Igusa local zeta functions of products of
linear polynomials, and their motivic and topological relatives. Our main
results include a self-reciprocity result for central arrangements defined over
fields of characteristic zero. We also prove combinatorial formulae for a
specialization of the flag Hilbert-Poincar\'e series for irreducible Coxeter
arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ in terms of
total partitions of the respective types. We show that a different
specialization of the flag Hilbert-Poincar\'e series, which we call the coarse
flag Hilbert-Poincar\'e series, exhibits intriguing nonnegativity features and
- in the case of Coxeter arrangements - connections with Eulerian polynomials.
For numerous classes and examples of hyperplane arrangements, we determine
their (coarse) flag Hilbert-Poincar\'e series. Some computations were aided by
a SageMath package we developed.
