  For a prime number p and a number field k, we first study certain etale
cohomology groups with coefficients associated to a p-adic Artin representation
of its Galois group, where we twist the coefficients using a modified Tate
twist with a p-adic index. We show that those groups are cofinitely generated
and explicitly compute an additive Euler characteristic. When k is totally real
and the representation is even, we relate the order of vanishing of the p-adic
L-function at a point of its domain and the corank of such a cohomology group
with a suitable p-adic twist. If the groups are finite, then the value of the
p-adic L-function is non-zero and its p-adic absolute value is related to a
multiplicative Euler characteristic. For a negative integer n (and for 0 in
certain cases), this gives a proof of a conjecture by Coates and Lichtenbaum,
and a short proof of the equivariant Tamagawa number conjecture for classical
L-functions that do not vanish at n. For p=2 our results involving p-adic
L-functions depend on a conjecture in Iwasawa theory.
