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So I was wondering if I can find some of the non trivial zeros of the Dirichlet-L function using mathematica. What I found out was there is a code ZetaZero[] which gives the non trivial zeros for the Riemann zeta function.

Is there a similar code for the Dirichlet-L function? I think not, as I wasn't able to find something like that. So, I am wondering how should I try to find them.

I tried plotting the graphs for some particular Dirichel-L functions, for example i tried to use

Plot[{Re[DirichletL[5, 3, 1/2 + I t]], Im[DirichletL[5, 3, 1/2 + I t]]}, {t, 0, 20}, PlotLegends -> "Expressions"]

From here I get a rough idea about the location of zeros on the line $\sigma=1/2$. But, I want the numerical values of $t$ which is where the graphs of both real and imaginary parts become zero in the graph.

Any ideas how should I go with this?

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$Version

(* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *)

Clear["Global`*"]

Plot[{Re[DirichletL[5, 3, 1/2 + I t]], Im[DirichletL[5, 3, 1/2 + I t]]}, {t, 
  0, 20}, PlotLegends -> "Expressions"]

enter image description here

Use FindRoot with starting values near the zero crossings from the Plot. Use Chop to remove the imaginary artifacts.

Chop[FindRoot[DirichletL[5, 3, 1/2 + I t] == 0, {t, #}, 
     WorkingPrecision -> 30] & /@ {7, 10, 12, 16, 18, 20}] // N

(* {{t -> 6.64845}, {t -> 9.83144}, {t -> 11.9588}, {t -> 16.0338}, {t -> 
   17.567}, {t -> 19.5407}} *)
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