  We study equivariant Arakelov-Euler characteristics of hermitian sheaves on
arithmetic varieties which support a tame action by a finite group G. The
tameness of the group action allows us to produce an equivariant Arakelov-Euler
characteristic in a particularly fine "projective" arithmetic class group. We
then show that the equivariant Arakelov-Euler characteristics of various
complexes of differentials determine the epsilon constants of the L-functions
of the motives obtained from the arithmetic variety using symplectic
representations of the group G. Our results may be viewed firstly as a higher
dimensional version of the Cassou-Nogu\`{e}s Taylor characterization of
symplectic Artin root numbers in terms of the hermitian structure of rings of
integers, and secondly as a signed equivariant version of Bloch's conductor
formula.
