  First-order relativistic conformal hydrodynamics in a general (hydrodynamic)
frame is characterized by a shear viscosity coefficient and two UV-regulator
parameters. Within a certain range of these parameters, the equilibrium is
stable and propagation is causal. In this work we study the correlation
functions of fluctuations in this theory. We first compute hydrodynamic
correlation functions in the linear response regime. Then we use the linear
response results to explore the analytical structure of response functions
beyond the linear response. A method is developed to numerically calculate the
branch cut structure from the well-known Landau equations. We apply our method
to the shear channel and find the branch cuts of a certain response function,
without computing the response function itself. We then solve the Landau
equations analytically and find the threshold singularities of the same
response function. Using these results, we achieve the leading singularity in
momentum space, by which, we find the long-time tail of the correlation
function. The results turn out to be in complete agreement with the loop
calculations in effective field theory.
