  For each central essential hyperplane arrangement $\mathcal{A}$ over an
algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the
Denef-Loeser motivic zeta function of $\mathcal{A}$. We prove a formula
expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of
related hyperplane arrangements. We use this formula to show that the map
taking each complex arrangement $\mathcal{A}$ to the Hodge-Deligne
specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the
realization space of any loop-free matroid. We also prove a combinatorial
formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of
the characteristic polynomials of related arrangements.
