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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. This formula shows that, in a precise sense, the degree to which 
$Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, 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.
