  Orders of vanishing of zeros of zeta functions have much arithmetic
information encoded in them. For the absolute zeta function, Dinesh Thakur gave
sufficient conditions for the order of vanishing of its zeros when the finite
field has two elements. Such conditions consider only principal ideals. This
result was generalized by Thakur and Diaz-Vargas. Now the conditions involve
not only the principal ideals but all the classes of ideals, still in the field
of two elements. In this work, we generalize these results to arbitrary finite
fields, using similar proofs of Thakur and Diaz-Vargas.
