It is well known that the Gauss sum of a Dirichlet character modulo $N$

$$G(\chi)=\sum_{a=1}^N\chi(a)e^{2\pi ia/N}$$

Moreover,$$\vert G(\chi)\vert=\sqrt{N}$$

when $\chi$ is primitive.

**Question** :If $G(\chi)=-\sqrt{N}$, the root number of L-function associated to the character $\chi$ is -1 or -i, which is not likely to occur. How to rule out such possibilities?

onlywhen $\chi = \overline{\chi}$, meaning $\chi$ is quadratic. For quadratic $\chi$ the root number is 1, so you learn nothing about a possible zero at $s=1/2$. And if $\chi$ is not quadratic then no value of the root number could force $L(1/2,\chi)$ to be $0$ since $\overline{\chi}$ is not the same character as $\chi$. $\endgroup$