Consider the polynomials $P_\ell(n)=\ell n^2+\ell n+d$. For level $\ell=1$, we see that the best polynomial of degree $2$ is $P_1(n)=n^2+n+41$, and this is the best of all. The discriminant of it is $\sqrt{-163}$. For levels $\ell=3$ and $\ell=2$, the best polynomials giving many primes are obviously (see the tables of arXiv:1807.07394 corresponding to z<0):
$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.