$\d x$ is the self-dual Haar measure with respect to $\psi_F$. For each $v\in \Sigma$, we set $\psi_{F_v}=\psi_{F}|_{F_v}$. For each $v\in\Sigma$, let $\d^\times x_v$ denote a Haar measure on $F_v^\times$ as ${\displaystyle \d^\times x_v=(1-q_v^{-1})^{-1}\, \tfrac{\d x_v}{|x|_v} }$ if $v<\inf$ and ${\displaystyle \d^\times x_v=\tfrac{\d x_v}{|x|_v} }$ if $v|\inf$. The idele norm $|\;|$ induces an isomorphism $\A^\times/\A^1\to\R_{>0}$. Choose the Haar measure $\d^\times x=\prod_{v\in\Sigma}\d^\times x_v$ on $\A^\times$ and normalize the Haar measure $\d^1 x$ on $\A^1$ in such a way that the quotient measure on $\R_{>0}$ is $\d t/t$, where $\d t$ is the Lebesgue measure on $\R$.

For a finite subset $S$ of $\Sigma$, we set $F_S=\prod_{v\in S}F_v$. Define the norm $|\; |_S$ on $F_S^\times$ by $|x|_S=\prod_{v\in S}|x_v|_v$ for $x=(x_v)\in F_S^\times$. Let $\psi_{F_S}(x)=\prod_{v\in S}\psi_{F_v}(x_v)$ for $x=(x_v)\in F_S$. For any vector space $\mathcal{V}$ over $F$, let $\cS(\mathcal{V}(F_S))$ and $\cS(\mathcal{V}(\A))$ denote the Schwartz spaces of $\mathcal{V}(F_S)$ and $\mathcal{V}(\A)$ respectively.

Global zeta functions and their principal parts
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Let $D$ be a quaternion algebra over $F$, set $D_\A:=D\otimes \A$. Then we have $V(\A)=D_\A\oplus D_\A$. Let $Z$ denote the center of $\GL_1(D)$. Choose an automorphic form $\phi_j\in L^2(Z(\A)D^\times\bsl D_\A^\times)$ $(j=1,2)$. Here, we refer to [@GJ p.145] for the sense of automorphic forms. Suppose that $\phi_1$ and 