sum rule (2) was not used. The other aim of this paper is to update the analysis such that all three sum rules can be used, and to employ the latest lattice QCD data including dynamical quarks as input for the spectral function fit.

The paper is organized as follows: In the next section, we introduce the quantities in quantum field theory that are necessary in our analysis, and explain how to derive the sum rules from the operator product expansion (OPE) and hydrodynamics [@Romatschke:2009ng]. Section \[sec:sumrule\] is devoted to the derivation of the sum rules in transverse and longitudinal channels, at small but finite spatial momentum. We also confirm that the spectral function evaluated at weak coupling and in the chiral limit satisfies these sum rules, and check to which energy region the sum rules are sensitive. We demonstrate that the sum rules can be used to improve the lattice QCD analysis for the zero-momentum case in Sec. \[sec:lattice\]. We summarize the paper and give concluding remarks in Sec. \[sec:summary\]. In the three appendixes, we evaluate the contributions to the spectral functions from the transport peak, the continuum, and the UV tail, at weak coupling and in the chiral 