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You can use Cauchy's theorem. Define the approximate zero of your function : zero = FindRoot[DirichletL[19, 10, s], {s, 0.5 + I}][[1, 2]] (* 0.5 + 1.51608 I *) Series will not consider this a pole of 1/DirichletL[19, 10, s] and I think this is why you get a zero residue. However, integrating on a small square around that pole one finds : Table[{eps, ...

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Inspired by tchronis's answer and Leonid's comment I came up with powers[int_] := FactorInteger[int][[All, 2]] perfectPowerQ[int_] := (GCD @@ powers[int] > 1) powers tells you what powers the prime factors are raised to to get the required number while perfectPowerQ checks that all these powers are the same.

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The function below returns the power of n. power[n_] := GCD @@ (FactorInteger[n][[All, 2]]) example: power[3111696] returns 4 because3111696=2^4*3^4*7^4 If you deal with very large numbers where FactorInteger simply can't help, you could use trial exponents IntegerQ[n^(1/m)] for m=2,3,4,.... Where to stop (I mean m) depends on the length of the ...

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