Computing $L$-functions with large conductor

An algorithm is given to efficiently compute $L$-functions with large conductor in a restricted range of the critical strip. Examples are included for about 21000 dihedral Galois representations with conductor near ${10}^7$. The data shows good agreement with a symplectic random matrix model. Also included are an elliptic curve $L$-function and one for a quadratic Dirichlet character.