  It is known that the numerical invariants Betti numbers and Bass numbers are
worthwhile tools for decoding a large amount of information about modules over
commutative rings. We highlight this fact, further, by establishing some
criteria for certain semidualizing complexes via their Betti and Bass numbers.
Two distinguished types of semidualizing complexes are the shifts of the
underlying rings and dualizing complexes. Let $C$ be a semidualizing complex
for an analytically irreducible local ring $R$ and set $n:=\sup C$ and
$d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if and
only if the $n$th Betti number of $C$ is one. Also, we show that $C$ is a
dualizing complex for $R$ if and only if the $d$th Bass number of $C$ is one.
