I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result: There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever $$ \sigma \geq 1 - \frac{c_1}{\log P}, $$ for all primitive characters $\chi$ modulus $q \leq P$, with the possible exception of at most one primitive character $\tilde{\chi} (mod \ \tilde{r})$. If it exists, the character $\tilde{\chi}$ is quadratic, and the unique exceptional real zero $\tilde{\beta}$ of $L(s, \tilde{\chi})$ satisfies $$ \frac{c_2}{\tilde{r}^{1/2} \log^2 \tilde{r}} \leq 1 - \tilde{\beta} \leq \frac{c_1}{\log P}. $$
I was wondering about the following.
1) Is there a lower bound for $\tilde{r}$?
2) I have seen in couple places without explanation that $(1 - \tilde{\beta}) \log P$ tends to $0$. How does one deduce to this?
Thank you very much.
PS I would like to change my question as the two questions have been answered in the comments. I would greatly appreciate if someone could explain me or provide reference for how to prove the inequality $\frac{c_2}{\tilde{r}^{1/2} \log^2 \tilde{r}} \leq 1 - \tilde{\beta}$ as I am having difficulty finding a reference for this.