I am not able to generalize the following equality involving Dirichlet series : $$(2\pi)^{-s}\Gamma(s)\left(\sum_{n=1}^{\infty}{\frac{c(n)}{n^s}}\right)=\int_{0}^{\infty}{\left(\sum_{n=1}^{\infty}{c(n)e^{-2\pi nt}}\right)t^{s-1}dt}$$ to algebraic number theory. In fact, I need a representation of $$\left(\frac{|\Delta_K|}{(2\pi)^n}\right)^s\Gamma_K(s)\left(\sum_{\mathfrak{a}}{\frac{c(\mathfrak{a})}{N\mathfrak{a}^s}}\right)$$ as an usual Mellin transform. In this formula, $K$ is an finite field extension over $\mathbb{Q}$ (degree $n$ and discriminant $\Delta_K$), the sum is over integral ideals in $K$ and $c(\mathfrak{a})$ is a complex number (we can suppose that the sum is absolutly converging for $\Re(s)$ large enough). $\Gamma_K$ is the 'higher dimensionnal $\Gamma$ function' defined (according to Neukirch 'Algebraic Number Theory') by $$\Gamma_K(s)=\int_{\textbf{R}_{+}^{*}}{N(e^{-y}y^s)\frac{dy}{y}}$$ (I don't really know if this function is an usual one but all the further definitions are in Neukirch).
I am begining with algebraic number theory (and furthermore I am a french user) so I first apologize for my english mistakes and please do not hesitate to detail the calculation...
Thanks a lot !