where $f_{gas}\equiv {{M_{\rm g}}}/M\equiv1/C_{M_g}$ is the gas fraction. Given the expression of the $M-{M_{\rm g}}$ and $M-T$ relations (Eqs. \[eq:mmg\] and \[eq:mt\]), the self-similar form for the $L-T$ relation is expressed as $$L = C_{LT}E(z)T^2, 
 \label{eq:lt}$$ and for the $L-M$ relation as $$L = C_{LM}E(z)^{7/3}M^{4/3}.
 \label{eq:lm}$$

Fitting Method
--------------

       Relation                         $X_0$                     $\beta$   $\gamma$
  ------------------ ------------------------------------------- --------- ----------
   $M-{M_{\rm g}}$        $2\times10^{13}\ M_\odot h^{-1}$           1         0
        $M-T$                     $3\ \textrm{keV}$                 3/2        -1
   $M-{Y_{\rm X}}$    $4\times10^{13}\ M_\odot h^{-1} \rm{keV}$     3/5       -2/5
   $L-{T_{\rm sl}}$               $3\ \textrm{keV}$                  2         1
        $L-M$             $5\times10^{14}\ M_\odot h^{-1}$          4/3       7/3

  : For each scaling relation (1st column): the pivot point, $X_0$ (2nd column), the self-similar slope, $\beta$ (3rd column), and the self-similar evolution parameter, $\gamma$ (4th column).

\[tabella1\]

To study the evolution of the parameters of the scaling relations, we fit them through a generic expression: $$\log F = \log C + \gamma \log E(z) + \beta \log (X/X_0).
   \label{11}
   $$ When $F$ represents the total mass, $M_{500}$, $X$ refers to one of the following mass-proxy: ${M_{\rm g}}$, ${T_{\rm mw}}$, ${T_{\rm sl}}$, ${Y_{\rm X}}$. Instead, when $F$ is the X-ray luminosity, $X$ stands for the spectroscopic-like temperature or the total mass. The value of each pivot point, $X_0$ is listed in 