For a number field $K$, let $\zeta_{ K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $\zeta_{ K}$ as well as its derivative $\zeta_{ K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^{ \gamma}$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].
Categories: math.NT