RSolve[{j0[n]==0.5*(1+Exp[I*2*n/3])*j0[n/3],j0[0]==1},j0,n]

j0[t_]:=(0.9793061030617947+0.16474594158442882*I) NProduct[(2/(1+(E^(2 I (3^(-1-K[1])))))),{K[1],1,-1-((Log[t])/(Log[3]))}];

j0[0]

j0[2]

j0[3^(-4)]

DiscretePlot[{Abs[j0[3^(n)]],Abs[j0appr[n]]},{n,-50,-1,1}]

j0appr[t_]:=(0.9793061030617947+0.16474594158442882*I) NProduct[(2/(1+(E^(2 I (3^(-1-K[1])))))),{K[1],1,IntegerPart[1+((Log[t])/(Log[3]))]}];

Plot[Abs[j0appr[t]],{t,0.005,100000}]

LogLogPlot[Abs[j0appr[t]],{t,0.005,10000000}]