We show that certain quadratic base change $L$-functions for $\mathrm{Gl}\left(2\right)$ are non-negative at their center of symmetry.

Let $\Gamma$ denote the modular group $PSL(2,Z)$. In this paper it is proved that ${\pi }_{\Gamma }\left(x\right)=\text{li}x+O\left(x^{\frac{71}{102}+ϵ}\right),ϵ\>0$. The exponent $\frac{71}{102}$ improves the exponent $\frac{7}{10}$ obtained by W. Z. Luo and P. Sarnak.