No one knows, but:
It is suspected that neither factoring nor discrete logarithm are NP-complete, but we have no proof. (Evidence for the suspicion: they are in NP $\cap$ coNP. See, e.g., https://cstheory.stackexchange.com/q/159/5038, https://cstheory.stackexchange.com/q/167/5038 for factoring. It's similarly easy to prove that discrete log is in NP $\...
Let's assume $X$ is NP-complete and $X$ is in co-NP.
We show that $NP \subseteq coNP$ and viceversa.
Because $X$ is NP-complete
$=>$ for each $L\in NP$ we can found a polytime function $f$ that $s\in L$ iff $f(s)\in X$.
But $X$ is in coNP
$=>$ for the polityme reduction closure of coNP, $L\in coNP$ too