Let $f \in \mathbb{C}[x_1,\ldots,x_n]$. Consider $D[s]$, where $D$ is the ring of polynomial coefficient differential operators in $n$ variables, and $s$ is an additional formal variable. Suppose $P(s)\in D[s]$ such that $$P(s)\cdot f^{s+1} = b(s)\, f^s,$$ for some $b(s)\in \mathbb{C}[s]$. Then the minimal, non-zero, monic $b(s)$ satisfying an equation as above is called the Bernstein-Sato polynomial or $b$-function of $f$.
It is well-known that the $b$-function always exists, and it is also well-known that it is related to the $D$-module $D[f^{-1}]$.
If $r$ is the minimal integral root of the $b$-function, then we see that $D[f^{-1}]$ is generated over $D$ by the single element $f^r$: notice that $$f^k = \frac{1}{b(k)}P(k)\cdot f^{k+1}$$ for any $k < r$ (because $b(k)\neq 0$), which means we can generate all powers of $f$ less than $r$. We can generate all powers of $f$ greater than $r$ by multiplying successively by $f$.
My question is, can we say anything further about either the $D$-module $D[f^{-1}]$ or $D(s)\,f^s$ from the minimal or maximal root of the $b$-function of $f$? Do the extremal roots of the $b$-function, or indeed any roots of the $b$-function, give information about any of the relevant $D$-modules?