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… 
The Selberg trace formula as a Dirichlet series
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