# The number of ideals of $\mathbb{Z}[x]$ containing $x(x-\alpha)(x-\beta)$ with given index

**Minisymposium:**
ASSOCIATION SCHEMES

**Content:**
Let $A$ denote an integral square matrix and $R$ the subring of the full matrix ring generated by $A$.
Then $R$ is a free $\mathbb{Z}$-module of finite rank, which guarantees that
there are only finitely many ideals of $R$ with given finite index.
Thus, the formal Dirichlet series $\zeta_A(s)=\sum_{n\geq 1}a_nn^{-s}$ is well-defined
where $a_n$ is the number of ideals of $R$ with index $n$.
In this talk we aim to find an explicit form of $\zeta_A(s)$ when
$A$ is the adjacency matrix of a graph whose eigenvalues
consists of exactly three integers.
which are integral.