# On exceptional eigenvalues of the Laplacian for $\Gamma_0(N)$

@article{Li2006OnEE,
title={On exceptional eigenvalues of the Laplacian for \$\Gamma\_0(N)\$},
author={Xian-jin Li},
journal={arXiv: Number Theory},
year={2006}
}
• Xian-jin Li
• Published 3 October 2006
• Mathematics
• arXiv: Number Theory
An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the non-Euclidean Laplacian for Hecke congruence subgroups $\Gamma_0(N)$ by the relation $\lambda_j=s_j(1-s_j)$ for $j=1,2,..., S$. Coefficients of the Dirichlet series involve all class numbers $h_d$ of real quadratic number fields. But, only the terms with $h_d\gg d^{1/2… 1 Citations The Selberg trace formula as a Dirichlet series • Mathematics • 2015 We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in ## References SHOWING 1-8 OF 8 REFERENCES Functorial products for$\mathrm{GL}_2 \times \mathrm{GL}_3$and the symmetric cube for$\mathrm{GL}_2\$
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