Is the n-th odd prime, which has 2 as nonresidue, polynomially bounded by n * ln(n)?

Question

We already know the $p_{n}$ prime is asymptotic with $n \times \ln (n)$. We already know from this book, the odd primes which have 2 as nonresidue are congruent 3 or 5 modulo 8. Is the odd prime $q_{n}$, which has 2 as nonresidue, polynomially bounded by $n \times \ln (n)$ or is asymptotic with another function?