Let $f$ be a polynomial in $n$ variables with complex coefficient. The $b$-function (or Bernstein-Sato polynomial) is the minimal nonzero monic polynomial $b_f(s)$, such that there exists differential operator $P(s) \in D_{\mathbb C^n}[s]$, satisfying $$ P(s) f^{s+1} = b_f(s) f^s. $$
On the other hand, for $\alpha \in \mathbb C$, we may consider the $D$-module, $$D_{\mathbb C^n} \cdot f^\alpha,$$ this is the $D_{\mathbb C^n}$-submodule generated by $f^\alpha$ inside $\mathbb C[z_1, \cdots, z_n][f^{-1}] \cdot f^\alpha$.
Question: is the following true? $$ \{\alpha \in \mathbb C \mid b_f(\alpha)=0 \} = \{\alpha \in \mathbb C \mid \frac{D_{\mathbb C^n} \cdot f^\alpha}{D_{\mathbb C^n} \cdot f^{\alpha+1}} \neq 0 \}. $$ The $ \supseteq$ direction is definitional.
Remark if we replace $\frac{D_{\mathbb C^n} \cdot f^\alpha}{D_{\mathbb C^n} \cdot f^{\alpha+1}}$ by $$\frac{D_{\mathbb C^n}[s] \cdot f^s}{D_{\mathbb C^n}[s] \cdot f^{s+1}} \otimes \frac{\mathbb C[s]}{(s-\alpha)}, $$then the above statement is true. See e.g. Kashiwara's book D-module and microlocal Analysis, section 6.5, lemma 6.21.
