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) and got a partial answer. I would like to see if there are more relevant results in this direction.

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) and got a partial answer. I would like to see if there are more relevant results in this direction.

Edit: note here that the $N$ in $L_N(\chi)$ is the same as the conductor of the character $\chi$. I forgot to emphasise this point in the first version, which led to some misunderstandings.