$L^1(\mathbb{R}/\mathbb{Z})$ norm of a trigonometric polynomial of Hardy-Littlewood

Question

For $x \in \mathbb{R}$ and $N \geq 1$ an integer, let $$S_N(x) = \sum_{n \leq N}n^{\frac{1}{2} + in}e^{2\pi i n x}. $$ I wonder if anyone knows what is the order of $$\int_{0}^{1}|S_N(x)|dx.$$ This polynomial was considered long ago by Hardy-Littlewood. It appears for instance in the book Trigonometric Series by Zygmund (if you have access to the book, it's in the 3rd edition, Vol 1, Chapter 5, Section 4, page 197). It follows from Theorem 4.7 there that $S_N(x) = O(N)$ uniformly in $x$. But does anyone know if one can do better in $L^1([0,1])$? Thanks!