A factor of integer polynomials with minimal integrals

For each positive integer $N$, let $S_N$ be the set of all polynomials $P(x) \in \Z[x]$ with degree less than $N$ and minimal positive integral over $[0,1]$. These polynomials are related to the distribution of prime numbers since $\int_0^1 P(x) \d x = \exp(-\psi(N))$, where $\psi$ is the second Chebyshev function. We prove that for any positive integer $N$ there exists $P(x) \in S_N$ such that $(x(1-x))^{\lfloor N / 3 \rfloor}$ divides $P(x)$ in $\Z[x]$. In fact, we show that the exponent $\lfloor N / 3 \rfloor$ cannot be improved. This result is analog to a previous of Aparicio concerning polynomials in $\Z[x]$ with minimal positive $L^\infty$ norm on $[0,1]$. Also, it is in some way a strengthening of a result of Bazzanella, who considered $x^{\lfloor N / 2 \rfloor}$ and $(1-x)^{\lfloor N / 2 \rfloor}$ instead of $(x(1-x))^{\lfloor N / 3 \rfloor}$.