= Solution
Let $|s\rangle=L_{-1}|\chi\rangle$ with $L_n|\chi\rangle=0$ for every $n\ge0$. Since $[L_0,L_{-1}]=L_{-1}$, $|s\rangle$ has $L_0$ eigenvalue one. Moreover,
$$
L_1|s\rangle=[L_1,L_{-1}]|\chi\rangle=2L_0|\chi\rangle=0,
$$
and for $n>1$,
$$
L_n|s\rangle=(n+1)L_{n-1}|\chi\rangle=0.
$$
It is therefore physical. It is a Virasoro descendant and hence spurious, while its norm is
$$
\langle s|s\rangle=\langle\chi|L_1L_{-1}|\chi\rangle
=2\langle\chi|L_0|\chi\rangle=0.
$$
Thus it is null.
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