  We explore some axioms of divergent series and their relations with conformal
field theory. As a consequence we obtain another way of calculating $L(0,\chi)$
and $L(-1,\chi)$ for $\chi$ being a Dirichlet character. We hope this
discussion is also of interest to physicists doing renormalization theory for a
reason indicated in the Introduction section. We introduce a twist of the
oscillator representation of the Virasoro algebra by a group of Dirichlet
characters and use this to give a 'physical interpretation' of why the values
of certain divergent series should be given by special L values. Furthermore,
we use this to show that some fractional powers which are crucial for some
infinite products to have peculiar modular transformation properties are
expressed explicitly by certain linear combinations of $L(-1, \chi)$'s for
appropriately chosen $\chi$'s, and can be understood physically as a kind of
'vacuum Casimir engergy' in our settings. We also note a relation between field
theory and our twisted operators. Lastly we give an attempt to reinterpret
Tate's thesis by a sort of conformal field theory on a number field.
