adopt the spectroscopic-like temperature in its expression: $$Y_{X} = {M_{\rm g}}\times {T_{\rm sl}}. 
    \label{eq:yx}$$ We stress, however, that in this new set of simulations, which includes the artificial diffusion term, [*the difference between the mass-weighted temperature and the spectroscopic-like one is largely reduced*]{} [see also @biffi.etal.2016] with respect to previous results [@rasia.etal.2014]. In the current $z=0$ AGN sample, the mean and standard deviation values for the ratio ${T_{\rm mw}}/{T_{\rm sl}}$ computed in the sphere of radius $R_{500}$ are equal to 1.02 and 0.10, respectively.

[**X-ray Luminosity.**]{} The bolometric luminosity is computed by summing up the contribution of the emissivity, $\epsilon_i$, of all gas particles within the sphere of radius $R_{500}$: $$L = \sum_{i}\epsilon_i= \sum_i n_{e,i}n_{H,i}\Lambda(T_i,Z_i)\Delta V_i, 
 \label{eq:lx}$$ where $n_{e,i}$, $n_{H,i}$ are number densities of electrons and hydrogen atoms, respectively, $\Delta V_i = m_i/\rho_i$ is the particle’s volume and $\Lambda_i$ is the interpolation of the cooling function pre-calculated in a fine grid of temperatures and metallicities. The cooling-function tables are created by adopting the APEC model [@2001ApJ...556L..91S] in XSPEC and by referring to the \[0.01-100\] keV energy band.

The Scaling Relations and the Self-Similar Prediction
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The total mass of a cluster can be related to the various ICM quantities presented 