In addition to the terms represented by the continuum OPE (14), there are discretization effects of O ( a $^{2}$Q$^{2}$) at finite lattice spacings. These can be eliminated by an extrapolation to the continuum limit, in principle. In our calculation obtained at a single lattice spacing, however, the error has to be carefully investigated. We use a lattice perturbation theory to estimate the discretization effects at large a $^{2}$Q2 regime as described below. We also note that the exact symmetries of the overlap fermion partly eliminate unphysical terms of O ( a $^{2}$Q$^{2}$) that violate the WT identities [4]. We therefore use (14) without including correction terms describing the discretization effects when we fit the lattice data of VPF extracted through (11) and (12). In our previous study in two-flavor QCD [4], we had to use more complicated method to extract the physical VPFs, because of non-conserved (axial-)vector currents. We now discuss a fit of the lattice VPF data to the OPE formula. In this analysis the renormalization scale is set to µ = 2 GeV when necessary, though the final result should not depend on µ up to higher order perturbative corrections. The gluon condensate 〈 ( α$_{s}$/π ) GG 〉 is defined only through the perturbative expression like (14) because of the renormalon ambiguity [21], hence we treat 〈 ( α$_{s}$/π ) GG 〉 as a free parameter to describe associated 1 /Q 4 corrections. On the other hand, the quark condensate 〈 ¯ qq 〉 is well-defined in the massless limit, as there is no mixing with lower dimensional operators because of the exact chiral symmetry of overlap fermion. Thus, the quark mass dependence of Π$_{V}$$_{+}$$_{A}$ |$_{OPE}$ ( Q $^{2}$, α$_{s}$ ), which comes from the third and fourth terms in (14), is given only as a function of α$_{s}$ , once the quark condensate is determined elsewhere. The running quark mass ¯ m ( Q ) is set to the value corresponding to the quark mass used in the lattice calculation. First, we obtain the value at 2 GeV using the non-perturbatively calculated Z -factor as ¯ m (2 GeV) = Z$_{m}$ (2 GeV) m$_{q}$ with Z$_{m}$ (2 GeV) = 0.806(12)(24)( + 0 − $_{11}$) [22]. Then, it is evolved to Q 2 using a three-loop running formula [19, 20]. For the quark condensate of up and down quarks, we use the value obtained in the recent analysis of the spectral density [23], i.e. 〈 ¯ qq 〉 = − [0.242(04)( +19 − $_{18}$) GeV]$^{3}$, which is defined in the MS scheme at 2 GeV. The strange quark condensate 〈 ¯ ss 〉 appears only as a contribution from sea quark and the associated coefficient C V + A ¯ ss ( Q $^{2}$, α$_{s}$ ) starts from O ( α$_{s}$ ). For the value of 〈 ¯ ss 〉 , we use the same value as the one of up and down quarks. In the fit of VPF using (14), there are three unknown parameters, α$_{s}$ , c and 〈 ( α$_{s}$/π ) GG 〉 . The QCD scale Λ (3) MS controls the running coupling constant α (3) s ( Q ), which is evaluated using the four-loop formula [24, 25]. 8