all the discussions on linear free divisors and related subjects we had over the last years and also for having done a good part of the computations used in this paper.

Linear Free Divisors and Gauß-Manin systems {#sec:LFDandGM}
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In this section, we first recall from [@dGMS] the construction of the family of Gauß-Manin systems associated to a linear section of a linear free divisor. We also give a more intrinsic definition of these Gauß-Manin systems as a direct image of a map constructed from two polynomials. Finally, we discuss the definition of the residue eigenvalues relevant for the present work, as introduced in [@dGMS].

We denote throughout this article by $V$ the complex vector space ${{\mathds C}}^n$.

1.  Let $D\subset V$ be a reduced hypersurface with defining equation $h$. Then $D$ is called a free divisor, if the sheaf $\Theta_V(-\log\,D):=\{\vartheta\in\Theta_V\,|\,\vartheta(h)\subset(h)\}$ is a free ${\mathcal{O}}_V$-module. If moreover a basis $(\xi_i)$ of $\Theta_V(-\log\, D)$ exists such that $\xi_i=\sum_{j=1}^n \xi_{ij}\partial_{x_j}$ where $\xi_{ij}$ are **linear** forms on $V$, then $D$ is called linear free.

2.  Let $G$ be the identity component of the algebraic group $G_D:=\left\{g\in\operatorname{Gl}(V)\,|\,g(D)\subset D\right\}$. Then $(V,G)$ is a pre-homogenous vector space in the sense of Sato (see, e.g., [@sk]), in 