Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.

Q1) Is it necessary that there always exists an $\mathcal{O}$ with class number one (e.g. Hurwitz order is of class number one in quaternions)? If it does, are they necessarily maximal? conversely, are the maximal orders (or Eichler orders i.e. intersection of two maximal orders) of class number one (e.g. Hurwitz quaternion)?

Q2) Do we know analytic continuation and functional equation (if exists) of Zeta functions over any $\mathbb{Z}$-order $\mathcal{O}$ (I am guessing a similar argument as in case of Epstein Zeta function could work).

In general, any reference on 'orders of Clifford algebras' would be highly helpful.

Thanks in advance.


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