$C$-parity in $\pi^0\pi^+\pi^-$ system I'm studying the conservation of the quantum number in the decay $\omega^0\rightarrow\pi^0\pi^+\pi^-$. Since
$P(\omega^0)=-1$
and
$P(\pi^0\pi^+\pi^-)=P(\pi^0)P(\pi^+)P(\pi^-)(-1)^{L_{+-}}(-1)^{L_{(+-)0}}=(-1)(-1)^{L_{+-}}(-1)^{L_{(+-)0}}$
To conserve parity $L_{+-}=L_{(+-)0}=1$
But to conserve C-parity
$C(\omega^0)=-1$
and
$C(\pi^0\pi^+\pi^-)=C(\pi^0)C(\pi^+\pi^-)(-1)^{L_{(+-)0}}=(-1)^{L_{(+-)0}}(-1)^{L_{+-}}$
since $C(\pi^0)=1$. Therefore if $L_{+-}=L_{(+-)0}=1$, C-parity is not conserved.
What am I missing?
 A: I think the problem with your would-be inconsistency (its a strong decay!) is the angular momentum factor you mysteriously inserted in the formula composing Cs.
Composing parity entails the spherical harmonics whose p-waves have negative parity, and they are two of them, as you soundly determined. (In fact, as a mental prop, you may consider the decay as a sequence, $\omega\to \rho^0\pi^0\to 3\pi$, the last step of which is also p-wave. Recall $J^{PC}: ~~~\omega ~~1^{--}; ~~ \rho^0 ~~1^{--}; ~~ \pi^0 ~~O^{-+}$.  )
But the composition of the two meaningful (neutral particle) Cs is
$$
C(\omega)= C(\rho^0)C(\pi^0)= (-)(+)= (-), 
$$
consistent. Recall, $C(\rho^0)\equiv C(\pi^+\pi^-)$, as the $\rho^0$ is a notional placeholder for the $\pi^+\pi^-$ p-wave eigenstate of C. There is no notional charge conjugation inserting the relative angular momentum $L_{(+-)0}$ between two charged particle constituents here.

Edit on comment : It is an XY problem... The WP point is correct for the C of a particle-antiparticle pair, and the $L_{+-}$ reminds you of the antisymmetry you already incorporated into the  $C=-$  of the placeholder $\rho^0$, which I put in for $\pi^+\pi^-$, to focus your thinking; nobody claimed you have to go through this channel; only that it easily summarizes the math.

*

*But there is no such  $(-)^L$ factor, as you wrongly inferred, for combining two neutral particles with well-defined C!

So, $L_{(+-)0}$ cannot and should not appear in the total C formula  that you invented out of whole cloth: it is pointless and wrong. Most good texts explain this.
