I've been trying to study the Dedekind zeta function, and there is one part I am having some trouble with. I am reading a paper with the derivation of the function equation which involves the integral: $\int_{G_{0}/V}^{}\prod_{k=1}^{n+1}\frac{dc_i}{c_i}$ where n is the rank of $\mathcal{O}_{K}^{×}$, $G_0$ is an $n+1$ tuple of nonnegative real numbers such that $\prod_{k=1}^{n+1}c_i^{N_i}$=1 ($N_i$=1 for $N_i\leq{r_1}$, 2 otherwise), and $V$ is the set of n+1 tuples $\left\{(|\sigma_i(u)|)\ |\ u\in\mathcal{O}_{K}^{×}\right\}$ (where $\sigma_i$ is a real/complex embedding of K), and ${G_{0}/V}$ is a fundamental domain for the quotient group. If we make the substitution $u_i=log(c_i)$, we get $\prod_{k=1}^{n+1}\frac{dc_i}{c_i}=dU$, so $\int_{log(G_{0}/V)}dU$ which simplifies the calculation.
We have that $-N_{n+1}u_{n+1}=N_{1}u_1+N_{2}u_2...N_{n}u_n$ (choosing $u_{n+1}$ arbitrarily, we could choose any other index), so this integral will be equal to the volume enclosed by each $u_i$ with $i<n+1$ since $u_{n+1}$ is dependent on the other free variables. The volume enclosed by these variables should be equal to the determinant of the matrix $(log|\sigma_{i}(u_j)|)$ where $i,j\leq n$ and $u_j$ is a set of generators for $\mathcal{O}_{K}^{×}$.
This makes sense to me because in $G_{0}/V$ we partition $G_0$ using $c \sim c'$ if $c_i=c'_i|\sigma_i(u)|$ for some unit $u$, after the substitution we get $u_i=u'_i+log(|\sigma_{i}(u)|)=u'_i+\sum_{m=1}^{n}h_mlog(|\sigma_{i}(u_m)|)$ (where each $h_i$ is some integer), taking the volume enclosed by these $u_i$ should be the fundamental domain of $\mathbb{R}^{n}/(log|\sigma_{i}(u_j)|)\mathbb{Z}^{n}$, which should just be the determinant of $(log|\sigma_{j}(u_i)|)$.
If I'm not mistaken, we also need to account for negative values each $u_i$ can take on so we would multiply this determinant by $2^n$. So overall, should we have $\int_{G_{0}/V}^{}\prod_{k=1}^{n+1}\frac{dc_i}{c_i}=2^ndet(log|\sigma_{j}(u_i)|)=2^{r_1-1}R_k$? (using the definition I find most often where $R_k=(N_jlog|\sigma_{j}(u_i)|)$ This makes sense to me but I am a complete novice so any help would be greatly appreciated.