Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$ 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.
 A: It is certainly possible to prove such bounds, for example by re-reading Iwaniec and Kowalski and being more careful and precise than them. There are no tricks and no shortcuts.
Such a computation would be messy. Since you are asking for conditional bounds, perhaps no one has bothered to work out the details. It would be take a long time, but not be too terribly difficult -- indeed it would be a great way to master the proofs.
I think Lior Silberman wrote up something along these lines (see the bottom of his website), but I was unable to get the link on his website to work. 
By the way, your use of the terminology `effective' is mistaken. The bounds you stated are effective --- "effective" means that one could work out the constant, not that anyone has actually done it. An example of an ineffective bound is the famous bound $h(-d) \gg d^{1/2 - \epsilon}$, where the implied constant depends on properties of a hypothetical Siegel zero -- so that this bound can't be used by itself to prove, e.g. that $h(-d) > 1$ for $d > 163$.
