Funder: Physics Division, National Center for Theoretical Sciences; doi: http://dx.doi.org/10.13039/100009638AbstractWe study a special running vacuum model (RVM) with Λ=3αH2+3βH04H−2+Λ0Λ=3αH2+3βH04H-2+Λ0, where αα, ββ and Λ0Λ0 are the model parameters and H is the Hubble one. This RVM has non-analytic background solutions for the energy densities of matter and radiation, which can only be evaluated numerically. From the analysis of the CMB power spectrum and baryon acoustic oscillation along with the prior of \alpha >0α>0 to avoid having a negative dark energy density, we find that \alpha <2.83\times 10^{-4}α<2.83×10-4 and β=(−0.2−4.5+3.9)×10−4β=(-0.2-4.5+3.9)×10-4 (95%% C.L.). We show that the RVM fits the cosmological data comparably to the ΛCDMΛCDM. In addition, we relate the fluctuation amplitude σ8σ8 to the neutrino mass sum ΣmνΣmν.</jats:p