Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet series
$$L_m(\chi)=L_m(1,\chi)=\sum_{n=1}^m\frac{\chi(n)}{n}.$$
It is well-known that $L_\infty(\chi)$ is positive, see chapter 6 of Apostol's 'Introduction to analytic number theory'. My question is about the partial sum $L_N(\chi)$. Here the subscript $N$ is the conductor of $\chi$. I have done some verification in sagemath, and for $N\le1000$, $L_N(\chi)$ is always positive. Is it always the case for any $N$? How to prove this? I asked this question in mathematics stack exchange([here](https://math.stackexchange.com/questions/4602090/positivity-of-partial-dirichlet-series-for-a-quadratic-character)) and got a partial answer. I would like to see if there are more relevant results in this direction.