Transfer function and characteristic equation

I have the following network (page 805, example 14.5, Agarwal & Lang, Foundations of Analog and Digital Circuits):

The associated transfer function for $i_z, v_z$ is $ H(s) = frac{I_z (s)}{V_z (s)} = frac{s^2 + (R/L) s + 1/(LC)}{s/C + R/(LC)}$. I can see how to get this from the impedance model of the circuit.

The example says that the numerator of this transfer function is the characteristic equation. The textbook example concludes that "the roots of the characteristic equation [numerator of transfer function] are complex and therefore the circuit is resonant", brackets comment mine.

I’m a bit confused by this since I thought that generally the characteristic equation is the denominator of the transfer function.

I think that the denominator of the transfer function is the characteristic equation of the output variable, so if you took the reciprocal of this transfer function, then $s^2 + (R/L) s + 1/(LC)$ would become the denominator and would be the characteristic equation of the new output variable $v_z$, so then you could analyze $s^2 + (R/L) s + 1/(LC)$ to find that voltage’s oscillation frequency $omega_0$, damping frequency $alpha $, quality factor Q, etc. However, these would not directly correspond to $i_z$ or the magnitude/phase plots of the original transfer function $ H(s) $. Is this correct?