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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):

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.

Consider the polynomials $P_\ell(n)=\ell n^2+\ell n+d$, where the values of $\ell$ and $d$ are taken from the tables of arXiv:1807.07394 corresponding to $z<0$. These polynomials have the special property that they give a prime number from $n=0$ to d-2. For level $\ell=1$, we see that the best polynomial of degree $2$ obtained from those tables is $P_1(n)=n^2+n+41$. The discriminant of it is $\sqrt{-163}$. For levels $\ell=3$ and $\ell=2$, the best polynomials giving many primes, and obtained from the corresponding tables are

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.

Consider the polynomials $P_\ell(n)=\ell n^2+\ell n+d$. For level $\ell=1$, we see that the best polynomial is $P_1(n)=n^2+n+41$. 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):

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.

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):

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.

Consider the polynomials $P_\ell(n)=\ell n^2+\ell n+41$. For level $\ell=1$, we see that $P_1(n)=n^2+n+41$. 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):

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.

Consider the polynomials $P_\ell(n)=\ell n^2+\ell n+d$. For level $\ell=1$, we see that the best polynomial is $P_1(n)=n^2+n+41$. 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):

$P_3(n)=3n^2+3n+23$, $\quad P_2(n)=2n^2+2n+19$.