We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$-functions of ${\mathrm{GL}}_d$ over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.

For a cocompact group $\Gamma$ of ${\text{SL}}_2\left(ℝ\right)$ we fix a real non-zero harmonic $1$-form $\alpha$. We study the asymptotics of the hyperbolic lattice-counting problem for $\Gamma$ under restrictions imposed by the modular symbols $〈\gamma ,\alpha 〉$. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.