I am reading a proof of Prime numbers in Dirichlet arithmetic progressions via this link: http://www.math.leidenuniv.nl/~evertse/ant14-7.pdf
However, according to his lemma 7.6, the writer wanted to make sure that $L(1+it,\chi)\ne0$ for all characters mod $q$. So he used theorem 5.5 which can be seen in this link http://www.math.leidenuniv.nl/~evertse/ant14-5.pdf) stating that $L(1+it,\chi) \ne 0$ for all $\chi^2\ne\chi_0$ where $\chi_0$ is the principal character mod $q$.
My problem is, for characters mod 4 which consist of $\chi_0, \chi_1$ , how he could make sure that $L(1+it,\chi_0)$ and $L(1+it,\chi_1)$ are not $0$? This is because $\chi_1^2 =\chi_0$. That means we cannot use theorem 5.5.
Historically, the case of real quadratic characters are the most difficult, and stumped many people for a very long time. Now there is a huge variety of known approaches, some even almost completely combinatorial.
In this document, you should just read one sentence further, when the author explains that he has yet to handle real quadratic characters. Then lemma 5.6 sets up theorem 5.7, which contains the proof for the real quadratic case.