A supplement to Chebotarev's density theorem

Let $L/K$ be a Galois extension of number fields with Galois group $G$. We show that if the density of prime ideals in $K$ that split totally in $L$ tends to $1/|G|$ with a power saving error term, then the density of prime ideals in $K$ whose Frobenius is a given conjugacy class $C\subset G$ tends to $|C|/|G|$ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of $\zeta_L(s)/\zeta_K(s)$.