# Analytic properties of Eisenstein series

Let $\Gamma$ be a discrete subgroup of $SL_2(\mathbb{R})$ which has a cusp at $\infty.$ suppose that $\mu(\Gamma\setminus\mathbb{H})<\infty,$ consider the Eisenstein series :$$E(z,s,\Gamma)=\sum_{\gamma\in\Gamma_\infty\setminus\Gamma}\dfrac{y^s}{|cz+d|^{2s}}$$ what is the analytic properties of $E(z,s,\Gamma)$ ?

• Look at the book "Elementary theory of Eisenstein series" by Tomio Kubota. – user1688 Mar 5 '17 at 15:22
• @Antonius, Thank you very much for the reference. – Med Mar 5 '17 at 15:33

The main properties of these Eisenstein series (meromorphic continuation, functional equation, poles and residues) are discussed and derived in Chapter 6 of Iwaniec: Spectral methods of automorphic forms (2nd edition, AMS, 2002). For their role in the spectral decomposition of $L^2(\Gamma\backslash\mathbb{H})$ see Chapter 7 of the same book. Further analytic properties in the arithmetic case (e.g. results about their value distribution) can be found in recent papers and theses (e.g. Luo-Sarnak, Spinu, Young, Huang-Xu).