L-functions.ipynb

Kernel: PARI/GP

f = mffrometaquo([1,1;47,1])

[[Vecsmall([4]), [47, 1, [[[47, [0]], [46, [46], [5]], [[47]~, Vecsmall([1])], [[47], [[46, [2, 1; 23, 1]]], Mat(1), [5], [46], [0]], Mat(1)], [23]~, 2, t - 1], y]], [Vecsmall([1, 47]), Vecsmall([1, 1])], 2]

a(M) = mfcoefs(f, M);

Lambda0(a, M, N, alpha, eta_f) = v = a(M+1); sum(n = 1, M, v[n+1] * (eint1(2*Pi*n*alpha) + I*eta_f/(2*Pi*n)*exp(-2*Pi*n/(N*alpha))))

(a,M,N,alpha,eta_f)->v=a(M+1);sum(n=1,M,v[n+1]*(eint1(2*Pi*n*alpha)+I*eta_f/(2*Pi*n)*exp(-2*Pi*n/(N*alpha))))

\p400

   realprecision = 404 significant digits (400 digits displayed)

N = 47;
eta_f = -I*sqrt(47);

z = Lambda0(a, 1000, N, 1/sqrt(N), eta_f)

0.09564042166229003918168887095103681547178295606049707957916733485374973581383481416294062462496175633698549894824445333730022991749336322530060501632457621071144249446536942287951851132774877436307098125608495894045465560650589772321615502514152294136759215873083150775289910468328593660422823709753308853040056827758811634051585485564710964596933368426357207137645041052672392946962565104804363471315

algdep(exp(z), 5)

x^5 + x^4 + x^3 - x^2 - 2*x - 1

polgalois(%)

[10, 1, 1, "D(5) = 5:2"]

kronecker(-47,5)

-1