"Source: Text/Geometry/Lseries.text";
"Line: 216";
"Date: Thu Sep 25 14:47:45 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Geometry/Lseries.text, line: 216
// Example: H139E4 ()
print "Example: H139E4";
ei := GetEchoInput();
SetEchoInput(true);
_<x> := PolynomialRing(Rationals());
K<s5> := NumberField( x^2-5 );
L<s205> := NumberField( x^2-205 );
C := Compositum(K,L);
e1 := C!(5+s5);
e2 := C!(41+s205);
E := ext<C | Polynomial( [ -e1*e2, 0, 1] )>;
A := AbsoluteField(E);
DefiningPolynomial(A);
Signature(A); // totally real
L := LSeries(A);
LCfRequired(L); // approx value, with old CFE
CFENew(L); // new vers of CheckFunctionalEquation
Evaluate(L, 1/2); // zero as expected
L`prod;
[ ComplexField(9)!Sign(x[1]) : x in $1 ];
Sign(L`prod[5][1]);
SetEchoInput(ei);
