  The Linear Independence hypothesis (LI), which states roughly that the
imaginary parts of the critical zeros of Dirichlet L-functions are linearly
independent over the rationals, is known to have interesting consequences in
the study of prime number races, as was pointed out by Rubinstein and Sarnak.
In this paper, we prove that a function field analogue of LI holds generically
within certain families of elliptic curve L-functions and their symmetric
powers. More precisely, for certain algebro-geometric families of elliptic
curves defined over the function field of a fixed curve over a finite field, we
give strong quantitative bounds for the number of elements in the family for
which the relevant L-functions have their zeros as linearly independent over
the rationals as possible.
