I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals $I_\mathfrak m$, trivial on $P_\mathfrak m^+$, the sum is over prime ideals $\mathfrak p$ of bounded norm. All I could find is $$\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p) = O(nx^{1/2}\ln(x)\ln(xd_KN\mathfrak m)),$$ for non-principal characters, and $$\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p) = x + O(nx^{1/2}\ln(x)\ln(xd_KN\mathfrak m)),$$ for the principal character, assuming GRH, from Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, p. 114. I would need to know more about the implied constant factor. Conditional bounds (GRH, ERH) are welcome.

I actually plan to use such a result to derive a bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)$, so any explicit bound on the later sum is also welcome.