[Math] Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields

Question

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, … of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the Dedekind zeta function of a quadratic number field. Is there any kind of algorithm which allows me to determine whether the number field is real or imaginary?

Answer 1

Not without an upper bound on the absolute value of the discriminant $\Delta$, because any finite list of $a_n$ amounts to a congruence condition on $\Delta$ that is satisfied by infinitely many $\Delta$ of either sign.