Let $\Gamma$ be a discrete subgroup of $\operatorname{SL}(2,\mathbb R)$, acting on the upper half-plane $\mathbb H$. Suppose that $\Gamma\backslash \mathbb H$ is non-compact and its compactification $\widetilde{\Gamma\backslash\mathbb H}$ has $n$ cusps. To each pair of cusps one can associate the corresponding real-analytic Eisenstein series. Let $\Phi(s)$ be the $n\times n$ matrix of these Eisenstein series. One can then define $$ \varphi(s)=\det\Phi(s). $$ This function is holomorphic at $s=\sigma+it$ for $\sigma>1$, satisfies the functional equation $\varphi(s)\varphi(1-s)=1$ and can be represented as follows $$ \varphi(s)=\left(\sqrt{\pi}\frac{\Gamma\left(s-\frac12\right)}{\Gamma(s)}\right)^n\sum_{j\geq 1}\frac{a_j}{b_j^{2s}} $$ for some $a_j\neq 0$ and $0<b_1<b_2<\ldots$

Is there anything known about $b_1=b_1(\Gamma)$ as a function of $\Gamma$? Is it continuous in any sense? Can one give a lower bound for $b_1(\Gamma)$ for a typical group $\Gamma$? I am particularly interested in the inequality $b_1(\Gamma)>4$, i.e. should we expect it to be true for a typical $\Gamma$.