zero of a polynomial is on the critical circle instead of critical line

(*Mathematica 8 start*)
a[n_] := Total[MoebiusMu[Divisors[n]]*Divisors[n]]
nn = 3*4*4;
log = Table[Sum[If[n == 1, 0, a[GCD[n, k]]/k], {k, 1, 6}], {n, 1, nn}]
n = nn;(*Increase n=500 for better precision*)
$MaxRootDegree = Round[Log[n]*n] + 1;
Monitor[z =
   Table[(x /.
       NSolve[Sum[(-1)^(k + 1)*x^Round[log[[k]]*n], {k, 1, n}] == 0,
        x])[[m]], {m, 1, nn}];, n]
ListPlot[Chop[
  Table[{Re[-nn*Log[z[[n]]]], Im[-4/Pi*Log[z[[n]]]]}, {n, 1,
    Length[z]}]]]
ListPlot[Chop[Table[{Re[z[[n]]], Im[z[[n]]]}, {n, 1, Length[z]}]],
 AspectRatio -> 4]
(*end*)

Is it any easier to determine if a zero of a polynomial is on the critical circle instead of a Riemann zeta zero on the critical line?
https://math.stackexchange.com/q/4297761/8530