In Phasor mode the dielectric constant is considered to be a complex number consisting:
ε = εrεo + j * σ/ω
where:
εr = relative permittivity
εo = permittivity of free space
σ = electrical conductivity
ω = angular frequency = 2πf
Also, j * σ/ω is the dielectric loss factor, ie complex permittivity in linear media and how it is related to the dielectric constant and the conductivity. In the restricted case of linear media, there are the Kramers-Kronig relations which expresses conductivity as an integral of dielectric constant over all frequencies, and also expresses dielectric constant as an integral of conductivity over all frequencies.
J = E*(σ + j*ε0*εr*ω), where J in A/m^2, E in V/m.
Re(J) is the "resistive" current density and Im(J) is the "capacitive" current density, when the E phasor has a real component only.
So the complex conductivity for a material is J/E = σ + j*ε0*εr*ω
You may know that both σ and εr are frequency dependent for many materials.
Are you really measuring j*ε0*εr*ω, or |σ + j*ε0*εr*ω|?