We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetrizable hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree and . We apply these asymptotic results to show that projections of the error are pointwise -superconvergent. We solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.