Why does NIntegrate give the wrong result for BesselJ

Question

Consider the following integral

z = 3*Exp[I*\[Theta]];
n = 20;
k = 1.25 \[Pi];
p = 1.5;

Integrate[BesselJ[(2 n + 1)*p, k*Abs[z]], {\[Theta], -\[Pi]/2, \[Pi]/2}]

However, when I replace Abs[z] with 3, it gives the correct answer:

Integrate[BesselJ[(2 n + 1)*p, k*3], {\[Theta], -\[Pi]/2, \[Pi]/2}]

Answer 1

$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

Clear["Global`*"]

It is a precision issue . Use exact values for the constants and use arbitrary-precision rather than machine precision.

z = 3*Exp[I*θ];
n = 20;
k = 5/4 π;
p = 3/2;

Block[{$MaxExtraPrecision = 100}, 
 Integrate[BesselJ[(2 n + 1)*p, k*Abs[z]], {θ, -π/2, π/2}] // N[#, 15] &]

(* 1.04339529077324*10^-37 *)