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I am fitting these data:

{{621826., 1.32918*10^46}, {1.65455*10^6, 
  6.40166*10^45}, {2.71762*10^6, 4.07986*10^45}, {3.78694*10^6, 
  2.97358*10^45}, {4.86065*10^6, 2.33453*10^45}, {5.93587*10^6, 
  1.91844*10^45}, {7.01104*10^6, 1.6266*10^45}, {8.08594*10^6, 
  1.4124*10^45}, {9.16232*10^6, 1.24839*10^45}}

with a function of the type: $$f(t)=\dfrac{a}{b+t}.$$ When I plot the result with the code:

ListPlot[vals,PlotStyle->Red,Epilog->{First@Plot[(1.1*10^52)/(2*10^3+t),{t,0,10^7},PlotRange->All,PlotLabel->"F1(t)"]}]

I expect to see a plot label $F1(t)$, but I do not see it. How do I add a plot label in this case?

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Put the PlotLabel inside the main plot, not inside Epilog:

ListPlot[vals, PlotStyle -> Red, PlotLabel -> "F1(t)", 
Epilog -> {First@Plot[(1.1*10^52)/(2*10^3 + t), {t, 0, 10^7}, PlotRange -> All]}]
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