[Wikipedia][1] states the time complexity of the General Number Field Sieve (GNFS) is
$$\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\log N)^{\frac{1}{3}}(\log \log N)^{\frac{2}{3}}\right),$$
where $N$ is the number to be factored, not the length of the input.

The same site also mentions [quasi-polynomial time][2] is
$$\mbox{QP} = \bigcup_{c \in \mathbb{N}} \mbox{DTIME} \left(2^{\log^c n}\right),$$
where $n$ is the length of the input, in other words, $n=\log N$.

Now my question is whether the GNFS is in quasi-polynomial time. It appears not, because of the existence of the $(\log N)^{1/3}$ exponent.

How about sub-exponential? [Wikipedia][3] says
$$\text{SUBEXP}=\bigcap_{\varepsilon>0} \text{DTIME}\left(2^{n^\varepsilon}\right).$$
I think the answer is still negative because $\varepsilon=1/3$ in this case and it can’t be smaller as in the intersection.

So the running time is exponential. Am I right?

  [1]: https://en.wikipedia.org/wiki/General_number_field_sieve
  [2]: https://en.wikipedia.org/wiki/Time_complexity#Quasi-polynomial_time
  [3]: https://en.wikipedia.org/wiki/Time_complexity#Sub-exponential_time