PDFs chosen for ZEUS-EW is $$\begin{aligned}
\label{eq:xgpar}
xg(x) &=   & A_g x^{B_g} (1-x)^{C_g} - A_g' x^{B_g'} (1-x)^{C_g'}  ,  \\
\label{eq:xuvpar}
xu_v(x) &=  & A_{u_v} x^{B_{u_v}}  (1-x)^{C_{u_v}}\left(1+E_{u_v}x^2 \right) , \\
\label{eq:xdvpar}
xd_v(x) &=  & A_{d_v} x^{B_{d_v}}  (1-x)^{C_{d_v}} , \\
\label{eq:xubarpar}
x\bar{U}(x) &=  & A_{\bar{U}} x^{B_{\bar{U}}} (1-x)^{C_{\bar{U}}} , \\
\label{eq:xdbarpar}
x\bar{D}(x) &= & A_{\bar{D}} x^{B_{\bar{D}}} (1-x)^{C_{\bar{D}}} .\end{aligned}$$

The normalisation parameters, $A_{u_v}, A_{d_v}, A_g$, are constrained by the quark-number sum rules and the momentum sum rule. The parameters $B_{\bar{U}}$ and $B_{\bar{D}}$ were replaced by a single $B$ parameter for the sea distributions. The strange-quark distribution is expressed as an $x$-independent fraction, $f_s$, of the $d$-type sea, $x\bar{s}= 0.4\, x\bar{D}$ at the starting scale. The parameter ${C_g'}$ is fixed to ${C_g'} = 25$[@Martin:2009iq]. The reduction to 13 parameters was implemented by replacing $x\bar{U}(x) =  A_{\bar{U}} x^{B_{\bar{U}}} (1-x)^{C_{\bar{U}}}\left(1+D_{\bar{U}}x\right)$ used for HERAPDF2.0 with Eq. \[eq:xubarpar\]. The reduction to 13 PDF parameters for ZEUS-EW greatly improved the stability of the fits necessary to determine the parameterisation uncertainties.

A 13-parameter fit with fixed SM $Z$ couplings, $\sin^2\theta_W$ and $M_W$, called ZEUS-13p, was performed as a reference. A fit with 13+4 parameters, called ZEUS-EW-Z, was used to extract the four couplings of the $Z$ to $u$- and $d$-type quarks. 