I will discuss joint work in progress with Will Sawin on function field multiple Dirichlet series constructed based on a set of axioms from algebraic geometry. These series have applications to moments of Dirichlet L-functions for characters of fixed order. We prove functional equations that fully determine many of these series and imply meromorphic continuation. The most striking feature of our work is a set of detailed predictions for the symmetries involved in any given moment computation. We find a complete tabulation of moments that are possible to compute via known multiple Dirichlet series methods, including several new moments