It is known that the local components of Dirichlet characters
of prime order l equidistribute on the unit circle as l
tends to
infinity taking prime values. Furthermore, the equidistribution
is locally independent. We show that both properties still hold
when restricting the Dirichlet characters to have their root num-
ber in any fixed interval in the unit circle. We show applications
of this result towards the linear statistics of the associated Dirichlet
L-functions.