The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function

where ${\displaystyle \mathbb {P} }$ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If ${\displaystyle \Gamma }$ is a subgroup of SL(2, R), the associated Selberg zeta function is defined as follows,

or

where p runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of ${\displaystyle \Gamma }$), and N(p) denotes ${\displaystyle \exp({\text{length of }}p)}$ (equivalently, the square of the bigger eigenvalue of p).

For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface.

The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface.

The zeros are at the following points:

The zeta-function also has poles at ${\displaystyle 1/2-\mathbb {N} }$, and can have zeros or poles at the points ${\displaystyle -\mathbb {N} }$.