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For a central, not necessarily reduced, hyperplane arrangement $f$ equippedwith any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing$f$, we consider a more general type of Bernstein--Sato ideal consisting of thepolynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying thefunctional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in\text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdotsf_{r}^{s_{r} + 1}.$  Generalizing techniques due to Maisonobe, we compute the zero locus of thestandard Bernstein--Sato ideal in the sense of Budur (i.e. $f^{\prime} = 1)$for any factorization of a free and reduced $f$ and for certain factorizationsof a non-reduced $f$. We also compute the roots of the Bernstein--Satopolynomial for any power of a free and reduced arrangement. If $f$ is tame, wegive a combinatorial formula for the roots lying in $[-1,0).$  For $f^{\prime} \neq 1$ and any factorization of a line arrangement, wecompute the zero locus of this ideal. For free and reduced arrangements oflarger rank, we compute the zero locus provided $\text{deg}(f^{\prime}) \leq 4$and give good estimates otherwise. Along the way we generalize a dualityformula for $\mathscr{D}_{X,\mathfrak{x}}[S]f^{\prime}f_{1}^{s_{1}} \cdotsf_{r}^{s_{r}}$ that was first proved by Narv\'aez-Macarro for $f$ reduced,$f^{\prime} = 1$, and $r = 1.$  As an application, we investigate the minimum number of hyperplanes one mustadd to a tame $f$ so that the resulting arrangement is free. This notion offreeing a divisor has been explicitly studied by Mond and Schulze, albeit notfor hyperplane arrangements. We show that small roots of the Bernstein--Satopolynomial of $f$ can force lower bounds for this number.

Combinatorially Determined Zeroes of Bernstein--Sato Ideals for Tame and Free Arrangements