I've read that, given two number fields $K, K'$ with equal Dedekind zeta functions $\zeta_K=\zeta_{K'}$ then, for every rational prime $p$, their local zeta functions are equal too, i.e. $\zeta_K^{(p)}=\zeta_{K'}^{(p)}$ where $$\zeta_K^{(p)}(s)=\prod_{\substack{ Q\in \mathcal{O}_K \\ Q|p }}\frac{1}{1-N(Q)^{-s}}.$$
Is there an easy way to see this?
In general, if a Dirichlet series $$L(s) = \sum_{n = 1}^\infty a(n)n^{-s}$$ has an Euler product $$L(s) = \prod_p L_p(s),$$ you can determine the local factor $L_p(s)$ by just reading off the coefficients of $L(s)$ at the prime powers: $$L_p(s) = \sum_{k=0}^\infty a(p^k) p^{-ks}.$$
This is true in particular for Dedekind zeta functions. Therefore, equal Dedekind zeta functions will have equal coefficents $a(n)$, therefore equal local zeta functions.