and $x\in{\mathbb{R}}$, we define the pair correlation function $F_\Phi$ by $$\begin{aligned}
F_\Phi(Q^{x},W)=\dfrac{1}{N_{\Phi}(Q)}\displaystyle\sum_{Q\leq q\leq 2Q}\dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi \ (\mathrm{mod} \ q)\\ \mathrm{primitive}}}\bigg|\displaystyle\sum_{\gamma_\chi}\mathcal{M}\Phi(i\gamma_\chi)Q^{i\gamma_\chi{x}}\bigg|^2.\end{aligned}$$ Using the asymptotic large sieve, Chandee, Lee, Liu and Radziwiłł[@CLLR] showed the following asymptotic formula under GRH $$\begin{aligned}
\label{FPhiasymp}
& F_{\Phi}(Q^{{x}},W)  \\ & = (1+o(1))\bigg[1-(1-|x|)_++\Phi\big(Q^{-|{x}|}\big)^2\log Q \bigg(\dfrac{1}{2\pi}\int_{-\infty}^{\infty}\big|\mathcal{M}\Phi(it)\big|^2{\,\text{\rm d}}t\bigg)^{-1}\bigg] + O\Big(\Phi(Q^{-|{x}|})\log^{1/2} Q\Big)\nonumber,\end{aligned}$$ which holds uniformly for $|{x}|\leq 2-\delta$ as $Q\to\infty$, for any fixed and sufficiently small $\delta>0$. Let $$N^*_{\Phi}(Q):=\displaystyle\sum_{Q\leq q\leq 2Q}\dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi \ (\mathrm{mod} \ q) \\ \text{primitive}}}\displaystyle\sum_{\gamma_\chi}m_{\rho_\chi}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2,$$ where $m_{\rho_\chi}$ denote the multiplicity of the nontrivial zero $\rho_\chi={\tfrac12}+i\gamma_\chi$ of $L(s,\chi)$. Since $$\displaystyle\sum_{\substack{\gamma_\chi\\\text{simple}}}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2\geq \displaystyle\sum_{\gamma_\chi}(2-m_{\rho_\chi})\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2$$ we obtain $$\begin{aligned}
 \label{ineqNstarwithN}
N_{\Phi,s}(Q)\geq 2N_{\Phi}(Q)-N^{*}_{\Phi}(Q).\end{aligned}$$ For any $g\in L^1({\mathbb{R}})$ with ${\widehat}g\in L^1({\mathbb{R}})$ we have the following explicit formula (Fourier inversion) $$\begin{aligned}
\sum_{Q\leq q\leq 2Q} \dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi \ (\mathrm{mod} \ q) \\ \text{primitive}}}\displaystyle\;\sum_{\gamma_\chi, \gamma'_\chi}\mathcal{M}\Phi(i\gamma_\chi)\mathcal{M}\Phi(i\gamma'_\chi)\,\widehat{g}\bigg(\dfrac{(\gamma_\chi-\gamma'_\chi)\log Q}{2\pi}\bigg)
     =N_{\Phi}(Q)\int_{-\infty}^{\infty}g(x)F_{\Phi}(Q^x,W){\,\text{\rm d}}x .\end{aligned}$$ Letting $f\in\mathcal{A}_{LP}$ and $g(x)=f(r(f)x/(2-\delta))$, for any primitive character $\chi  \ (\mathrm{mod} \ q)$ we obtain $$\begin{aligned}
&\sum_{\gamma_\chi, \gamma'_\chi}\mathcal{M}\Phi(i\gamma_\chi)\mathcal{M}\Phi(i\gamma'_\chi)\,\widehat{g}\bigg(\dfrac{(\gamma_\chi-\gamma'_\chi)\log Q}{2\pi}\bigg) \\ & = \displaystyle\sum_{\gamma_\chi}m_{\rho_\chi}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2\,\widehat{g}(0) \nonumber + \displaystyle\sum_{\gamma_\chi\neq \gamma'_\chi}\mathcal{M}\Phi(i\gamma_\chi)\mathcal{M}\Phi(i\gamma'_\chi)\,\widehat{g}\bigg(\dfrac{(\gamma_\chi-\gamma'_\chi)\log Q}{2\pi}\bigg) \nonumber \\
        & \geq \dfrac{2-\delta}{r(f)}\,\displaystyle\sum_{\gamma_\chi}m_{\rho_\chi}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2. 
        \end{aligned}$$ This implies that $$\begin{aligned}
& \sum_{Q\leq q\leq 2Q} \dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi \ (\mathrm{mod} \ q) \\ \text{primitive}}}\displaystyle\;\sum_{\gamma_\chi, \gamma'_\chi}\mathcal{M}\Phi(i\gamma_\chi)\mathcal{M}\Phi(i\gamma'_\chi)g\bigg(\dfrac{(\gamma_\chi-\gamma'_\chi)\log Q}{2\pi}\bigg)  \geq \dfrac{2-\delta}{r(f)}N^*_{\Phi}(Q).\end{aligned}$$ On the other hand, observing that $$\Phi\big(Q^{-|{x}|}\big)^2\log Q \bigg(\dfrac{1}{2\pi}\int_{-\infty}^{\infty}\big|\mathcal{M}\Phi(it)\big|^2{\,\text{\rm d}}t\Big)\to {{\boldsymbol}\delta}(x),$$ as $Q\to \infty$ (in the distributional sense) and that 