Taylor expansion of Dedekind zeta function at $s=0$.

Question

Let $K$ be a number field and let $\zeta_K(s)$ be the corresponding Dedekind zeta function. I want to show that the order of $\zeta_K(s)$ at $s=0$ equals $r$, where $r$ is the rank of the unit group of $K$ and also that the leading coefficient equals $\frac{-hR}{w}$, where $h$ is the class number of $K$ , $R$ is the regulator of $K$ and $w$ is the number of roots of unity in $K$. I know that this should be straightforward to prove using Dirichlet's class number formula and the functional equation for the completed zeta function $\Lambda_K(s)$. My attempt was to start with $\Lambda_K(s)=\Lambda_K(1-s)$, multiply both sides with $s^r$, take the limit as s tends to zero and simplify using Dirichlet's class number formula and the fact that the Gamma function has a simple pole at $s=0$ with residue $1$. However, I am not quite succeeding since I am getting an extra factor of $2^{r_1}$ without the negative sign. Could anyone please help me out by showing me the correct calculation? Thank you.