  We describe a procedure for determining the existence, or non-existence, of
an algebraic variety of a given conductor via an analytic calculation involving
L-functions. The procedure assumes that the Hasse-Weil L-function of the
variety satisfies its conjectured functional equation, but there is no
assumption of an associated automorphic object or Galois representation. We
demonstrate the method by finding the Hasse-Weil L-functions of all
hyperelliptic curves of conductor less than 500.
