Abstract: Given a holomorphic function f, an analytic way of understanding its singularity originates from the famous Gelfand’s problem, where he proposed in his 1954 ICM talk to study the distribution |f|^2s, where s is a holomorphic parameter, and asked for its meromorphic extension. This distribution is the Archimedean analogue of the Riemann zeta function. In 1972, Bernstein solved this problem by introducing the Bernstein-Sato polynomial, which is one of the origins of the theory of algebraic D-modules. The poles of this distribution remained mysterious. In this talk, I will report on recent joint work with Dougal Davis and András Lőrincz, where we prove that the largest nontrivial pole must be the largest nontrivial root of the Bernstein-Sato polynomial and determine the pole order in terms of the multiplicity of the corresponding root. This resolves a question of Loeser and Mustață-Popa in a strong sense. One of the key input is the theory of complex Hodge modules of Sabbah and Schnell, a Hodge-theoretic enrichment of algebraic D-modules. Time permitting, I will discuss other poles of the zeta function as well as an analytic characterization of the Kashiwara-Malgrange V-filtration.