We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called Flag Hilbert-Poincaré series. These series are closely related to the Igusa local zeta functions of the product of linear polynomials, and their motives and topological relationships. Our main results include self-reciprocal results for centered configurations defined in fields with zero properties. We also prove combinatorial formulas for specializations of the Flag Hilbert-Poincare series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$. of total partitions of each type. We show that another specialization of the Flag Hilbert-Poincare series, called the coarse Flag Hilbert-Poincare series, exhibits interesting non-negative features and, in the case of the Coxeter configuration, a connection with the Euler polynomials. For a number of classes and examples of hyperplane configurations, we determine their (coarse) Flag Hilbert-Poincare series. Some computations were assisted by the SageMath package we developed.
