Ten years old question asks about Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $K$ be the number field with the degree 24 defining polynomial
x^24 + 3*x^23 - 2*x^22 - 43*x^21 + 81*x^20 + 1579*x^19 + 2434*x^18 - 5192*x^17 + 4678*x^16 - 41425*x^15 + 423527*x^14 + 1352722*x^13 + 5199537*x^12 - 13364304*x^11 - 138065100*x^10 + 228783352*x^9 + 1254448448*x^8 - 3179566016*x^7 + 4205123840*x^6 + 139822208*x^5 - 31439415040*x^4 + 28607489536*x^3 + 330701977600*x^2 - 807251576832*x + 635017424896
and $\zeta_K$ be the zeta function of $K$.
According to computations with Magma online $\zeta_K$ has double zero at $1/2$, but this could be numerical instability with low precision.
Q1 Does $\zeta_K$ really have double zero at $1/2$?
Q2 Does the double zero implies some properties of $K$?
Q3 Is there construction of number fields which have double or higher order zeros of zeta function at $1/2$?
In comment, David E Speyer suggested $L(s,\chi)^2$ should divide $\zeta_K$.
We believe that all zeros of $L(s,\chi)$ will be double zeros of $\zeta_K$.
Checking the first three zeros of the linked question (they are with very low precision) give $|\zeta_K(\rho)| \sim 10^{-22}$ and $|\zeta_K'(\rho)| \sim 10^{-11}$. This numerical results appear to suggest that the first three complex zeros are double.
Q4 Are there other, preferably infinitely many, double zeros of $\zeta_K$?
Magma online code in case someone is interested:
Z1<x>:=PolynomialRing(Integers());
p1:=x^24 + 3*x^23 - 2*x^22 - 43*x^21 + 81*x^20 + 1579*x^19 + 2434*x^18 - 5192*x^17 + 4678*x^16 - 41425*x^15 + 423527*x^14 + 1352722*x^13 + 5199537*x^12 - 13364304*x^11 - 138065100*x^10 + 228783352*x^9 + 1254448448*x^8 - 3179566016*x^7 + 4205123840*x^6 + 139822208*x^5 - 31439415040*x^4 + 28607489536*x^3 + 330701977600*x^2 - 807251576832*x + 635017424896;
Nf<w>:=NumberField(p1);
L:=LSeries(Nf);
I:=Sqrt(-1);
//t0:=0.99014365233;
//t0:=1.38830360231;
t0:=2.35103235859;
A:=1/2+I*t0;
Evaluate(L,A);
Evaluate(L,A : Derivative:=1);