Numerical tests of modularity

We propose some numerical tests for identifying L-functions of automorphic representations of GL(r) over a number field. We then apply the tests to various conjectured automorphic L-functions, provid- ing evidence for their modularity and the associated Riemann hypotheses. Our chief examples are the Hasse-Weil L-functions attached to curves of genus 2 over Q and to elliptic curves over Q( √ −1). We discuss also three miscellaneous applications. The first two include the L-functions of high symmetric powers of Ramanujan'sand the modular form in S2(� 0(11)). The third application is an even 2-dimensional icosahedral Galois repre- sentation over Q, which conjecturally corresponds to a Maass form of eigenvalue 1 .