4. In the Cartesian coordinate system, the hyperbola $\Gamma: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$, a line with an inclination angle of $\frac{\pi}{4}$ passes through one vertex of $\Gamma$ and another point $(2,3)$ on $\Gamma$, then the eccentricity of $\Gamma$ is $\qquad$ .