The zeta function of a mathematical operator \mathcal O is a function defined as for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace. The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues \lambda_i of the operator \mathcal O by It is used in giving a rigorous definition to the functional determinant of an operator, which is given by The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.