On the Northcott property of zeta functions over function fields

Pazuki and Pengo defined a Northcott property for special values of zeta functions of number fields and certain motivic L-functions. We determine the values for which the Northcott property holds over function fields with constant field F_q outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably [Formula presented] and for s real such that 1/2≤s≤1, and we obtain a partial result for complex s in the case 1/2<Re(s)≤1 using recent advances on the Shifted Moments Conjecture over function fields.