Abstract:
We construct some regions without zeros of the confluent hypergeometric function $_1F_1(-n;d;z)$ ($n\in\mathbf N, d\in\mathbf C$). The main result is as follows. If
$$
-n+\frac7{16}\geq\operatorname{Re}(d),\quad_1F_1(-n;d;z)=0,
$$
then
$$
-\operatorname{Re}(d)-\operatorname{Im}(d)\operatorname{tg}\biggl(\frac{\arg(z)}2\biggr)
\geq|z|>\operatorname{Re}(z)+2\biggl(-n+\frac7{16}-
\operatorname{Re}(d)\biggr).
$$