next section to obtain the Euler product of the contribution of each orbit to the zeta function.

Global zeta functions with toric periods {#sec:globalzeta424}
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Notations {#sec:notation}
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Let $F$ be an algebraic number field, and let $\Sigma$ denote the set of all the places of $F$. For any $v\in\Sigma$, we denote by $F_v$ the completion of $F$ at $v$. If $v<\inf$, we write $\fo_v$ for the ring of integers of $F_v$, and $\varpi_v$ for a prime element of $\fo_v$. We put $q_v=\#(\fo_v/\varpi_v\fo_v)$. Let $\A$ denote the adele ring of $F$. Let $\d x$ denote the Haar measure on $\A$ normalized by $\int_{\A/F} \d x=1$. For each $v<\inf$, we fix a Haar measure $\d x_v$ on $F_v$ normalized by $\int_{\fo_v}\d x_v=1$. Further, we choose Haar measures $\d x_v$ on $F_v$ for $v|\inf$ so that $\d x = \Delta_F^{-1/2} \prod_{v\in\Sigma} \d x_v$ holds, where $\Delta_F$ denotes the absolute discriminant of $F/\Q$.

We denote by $|\; |_v$ the normal valuation of $F_v$. Then, we have $\d (ax_v)=|a|_v \d x_v$ for any $a\in F_v^\times$. Define the idele norm $|\; |=|\; |_\A$ on $\A^\times$ by $|x|=|x|_\A=\prod_{v\in\Sigma} |x_v|_v$ for all $x=(x_v)\in\A^\times$. Choose a non-trivial additive character $\psi_\Q$ on $\A_\Q/\Q$, and set $\psi_F=\psi_\Q \circ \Tr_{F/\Q}$. Then, 