Relations between different zeta functions for a simple algebra I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether this adjective is proper; I just followed Voight's famous book: Quaternion algebras) zeta functions for simple algebras (over a number field).
Meanwhile, I learnt basic ideas and calculations of the "modern" Godement-Jacquet theory (their most famous monograph: Zeta functions for simple algebras).
So what made me puzzled is their relations. In modern Godement-Jacquet theory, the so called "standard" way is to use harmonic analysis on adelic groups (global theory) to define the zeta integral. So we denote by $G$ the group of units of the simple algebra, and define:
$$
Z(\Phi, s, \varphi)=\int_{G_{\mathbb{A}}} \Phi(g)|\operatorname{det} g|^{s} \varphi(g) \mathrm{d} g
$$
where $\Phi$ is taken to be a Schwartz function and $\phi$ a matrix coefficient of an automorphic representation. Then we need some work to obtain the $L$-factor and $\gamma$-factor, etc, which established its automorphic $L$-function.
On the other hand, for example (for simplicity), let $B$ be a quaternion algebra over a number field, $O\subset B$ a maximal order. Then there's also a definition of the "classical" zeta function:
$$
\zeta_{O}(s):=\sum_{I \subseteq O} \frac{1}{\mathrm{~N}(I)^{s}}=\sum_{\mathfrak{n}} \frac{a_{\mathfrak{n}}(O)}{\mathrm{N}(\mathfrak{n})^{2 s}}
$$
where $a_{\mathfrak{n}}(O):=\#\{I \subseteq O: \operatorname{nrd}(I)=\mathfrak{n}\}$ (see Voight's book, section 26.3). This could be generalised to general simple algebras.
So I want to ask: is their any relation between these two "zeta" functions? I think there must be some: when I was checking Tamagawa's annals paper, I found he mentioned Godement's Bourbaki's report on this, and he said:
"the ordinary $\zeta$-function of a division algebra was introduced by K.Hey, and generalized by M.Eichler to $L$-functions with abelian characters. The first attempt to generalize these theories to $\zeta$-functions with non-abelian characters is due to H.Maass. Later, R.Godement gave a method to get the most general formulations on these matters. In this note, we will define a type of $\zeta$-functions (maybe that's my question: the "calssical" ones) of a division algebra over a number field which are included in Godement's work as a special case, and for which one can develop the theory of Euler products. The latter theory has its own meaning as an application of the theory of spherical functions on $p$-adic algebraic groups.   ......"
Could any expert give me some help/advice? Thanks in advance!! @Paul Garrett
 A: Let's recall some history about $L$-functions not on division algebras first.  Dirichlet introduced $L$-functions of certain characters, later extended by Hecke.  These characters may be viewed as automorphic representations of GL(1) in Langlands' paradigm.  For the trivial character over $\mathbb Q$, you get the Riemann zeta function, and for the trivial character over a number field $K$, you get the Dedekind zeta function for $K$.
Later, $L$-functions were attached to higher-dimensional objects by Hecke, Maass, Artin, Langlands, ...  E.g., $L$-functions of modular forms or 2-dimensional complex Galois representations should correspond to automorphic representations of GL(2) in the Langlands paradigm.
Now modular forms also correspond to automorphic forms on quaternion algebras, and one can directly try to construct $L$-functions on quaternion algebras $B$.  This is the degree 2 case of what Godement-Jacquet do---they construct zeta ($L$) functions associated to representations of $B$, which in the holomorphic case correspond to $L$-functions of classical modular forms.  However, if you take the trivial representation on $B^\times$ (giving the classical zeta function of $B$), this corresponds to an Eisenstein series in the language of modular forms.  The fact that these are equal amounts to the fact that you can write the classical zeta function of $B$ in terms of the Riemann/Dedekind zeta function.
