In two papers Deninger proved the following:
- If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$, subject to the usual multiplication rules, $\Theta:e^{\alpha}\longmapsto\alpha e^{\alpha}$ is the usual derivation and $\mathbb{L}=\mathbb{C}[\log_{q}(1)]\subseteq B$, the $\zeta$-factor at $q$ is given by \begin{equation*} \det\left(\frac{\log(q)}{2\pi i}(s-\Theta)\mid\mathbb{L}_{\lambda}\right)^{-1}=\frac{1}{1-\lambda q^{-s}} \end{equation*} where $\mathbb{L}_{\lambda}$ is the $\lambda^{-1}$-eigenspace of $\Theta$.
- At the infinite primes the $\zeta$-factor is given by \begin{equation*} \det\left(\frac{1}{2\pi}(s-\Theta)\right)^{-1} \end{equation*} where $\Theta$ is not the usual derivation, but rather defined via $\Theta(f)=T\frac{d}{dT}f$.
I am rather puzzled by the structural difference between these two cases, in particular by the occurrence of the factor $\frac{\log q}{i}$ in the former result. Where - philosophically speaking - does that factor come from? Is there an intrinsic reason why it has to appear, other than the mere Well, that's the way it is?
Put differently: Why is $\frac{\log(\infty)}{i}=1$?
Deninger, Christopher, On the (\Gamma)-factors attached to motives, Invent. Math. 104, No. 2, 245-261 (1991). ZBL0739.14010.
Deninger, Christopher, Local (L)-factors of motives and regularized determinants, Invent. Math. 107, No. 1, 135-150 (1992). ZBL0762.14015.
