  We give a presentation via generators and relations of the local graded
paramodular Hecke algebra of prime level. In particular, we prove that the
paramodular Hecke algebra is isomorphic to the quotient of the free
$\mathbb{Z}$-algebra generated by four non-commuting variables by an ideal
generated by seven relations. Using this description, we derive rationality
results at the level of characters and give a characterization of the center of
the Hecke algebra. Underlying our results are explicit formulas for the product
of any generator with any double coset.
