Let $L$ be a number field and let $\zeta\_L(s)$ be its associated Dedekind zeta function. It is known that $\zeta\_L(s)$ has at most one zero in the region
$$1 - \frac1{4 \log d\_L} \leq \sigma \leq 1, \qquad |t| \leq \frac1{4\log d\_L},$$
(as usual, $s=\sigma + it$) where $d\_L$ is the discriminant of $L / \mathbb{Q}$. If such a zero exists, then it is called *Stark zero* of $L$, and it is known that it must be real and simple.


The Stark zero of $L$ appears in the error term of the unconditional form of Chebotarev theorem given by Lagarias and Odlyzko [1]. In particular, if $L$ has no Stark zero, then the error term is smaller.


**My question is:** For which number field $L$ we know that the Stark zero does not exists? I mean both general theorems and concrete examples (e.g., is there a table of quadratic fields with no stark zero?)


[1] J. Lagarias and A. Odlyzko. Effective versions of the Chebotarev density theorem. Algebraic Number Fields (A. Frohlich, editor), Academic Press, New York, 1977, pp. 409-464



For which number fields we know the nonexistence of Stark zeros?
In [Real zeros of real odd Dirichlet $L$-functions](https://www.ams.org/journals/mcom/2004-73-245/S0025-5718-03-01537-0/S0025-5718-03-01537-0.pdf), Mark Watkins showed in 2003 that $L(s,\chi\_d)$ (as in the title) has no positive real zeros for $d<300,000,000$. (I think this is still state of the art.). This answers the questions for the corresponding complex quadratic fields $\mathbb Q(\sqrt{-d})$.


