void
acb_dirichlet_l(acb_t res, const acb_t s,
const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
{
if (!acb_is_finite(s))
{
acb_indeterminate(res);
}
else if (G == NULL || G->q == 1)
{
acb_dirichlet_zeta(res, s, prec);
}
else if (dirichlet_char_is_primitive(G, chi) &&
(arf_cmp_d(arb_midref(acb_realref(s)), -0.5) < 0 ||
(G->q != 1 && dirichlet_parity_char(G, chi) == 0 &&
arf_cmpabs_d(arb_midref(acb_imagref(s)), 0.125) < 0 &&
arf_cmp_d(arb_midref(acb_realref(s)), 0.125) < 0)))
{
/* use functional equation */
acb_t t, u, v;
int parity;
ulong q;
parity = dirichlet_parity_char(G, chi);
q = G->q;
acb_init(t);
acb_init(u);
acb_init(v);
/* gamma((1-s+p)/2) / gamma((s+p)/2) */
acb_add_ui(t, s, parity, prec);
acb_mul_2exp_si(t, t, -1);
acb_rgamma(t, t, prec);
if (!acb_is_zero(t)) /* assumes q != 1 when s = 0 */
{
acb_neg(u, s);
acb_add_ui(u, u, 1 + parity, prec);
acb_mul_2exp_si(u, u, -1);
acb_gamma(u, u, prec);
acb_mul(t, t, u, prec);
/* epsilon */
acb_dirichlet_root_number(u, G, chi, prec);
acb_mul(t, t, u, prec);
/* (pi/q)^(s-1/2) */
acb_const_pi(u, prec);
acb_div_ui(u, u, q, prec);
acb_set_d(v, -0.5);
acb_add(v, v, s, prec);
acb_pow(u, u, v, prec);
acb_mul(t, t, u, prec);
acb_sub_ui(u, s, 1, prec);
acb_neg(u, u);
acb_conj(u, u);
acb_dirichlet_l_general(u, u, G, chi, prec);
acb_conj(u, u);
acb_mul(t, t, u, prec);
if (dirichlet_char_is_real(G, chi) && acb_is_real(s))
arb_zero(acb_imagref(t));
}
acb_set(res, t);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
else
{
acb_dirichlet_l_general(res, s, G, chi, prec);
}
}
int main()
{
slong iter, bits;
flint_rand_t state;
flint_printf("properties....");
fflush(stdout);
flint_randinit(state);
for (bits = 5; bits <= 30; bits += 5)
{
for (iter = 0; iter < 50; iter++)
{
dirichlet_group_t G;
dirichlet_char_t chi, psi;
ulong q, iter2;
q = 2 + n_randint(state, 1 << bits);
dirichlet_group_init(G, q);
dirichlet_char_init(chi, G);
dirichlet_char_init(psi, G);
/* check number char properties */
for (iter2 = 0; iter2 < 100; iter2++)
{
ulong m, n;
ulong p1, p2, pairing, cm, cn, q2, q3;
dirichlet_group_t G2, G3;
dirichlet_char_t chi2, chi3;
if (iter2 == 50)
dirichlet_group_dlog_precompute(G, 5);
/* one random character */
do
m = n_randint(state, q);
while (n_gcd(q, m) > 1);
dirichlet_char_log(chi, G, m);
p1 = dirichlet_order_ui(G, m);
p2 = dirichlet_order_char(G, chi);
check_eq(p1, p2, q, m, "order m", "order chi");
p1 = dirichlet_conductor_ui(G, m);
p2 = dirichlet_conductor_char(G, chi);
check_eq(p1, p2, q, m, "conductor m", "conductor chi");
p1 = dirichlet_parity_ui(G, m);
p2 = dirichlet_parity_char(G, chi);
check_eq(p1, p2, q, m, "parity m", "parity chi");
p1 = dirichlet_char_is_real(G, chi);
p2 = (dirichlet_order_char(G, chi) <= 2);
check_eq(p1, p2, q, m, "is_real", "(order <= 2)");
/* check index */
p1 = dirichlet_index_char(G, chi);
dirichlet_char_index(psi, G, p1);
if (!dirichlet_char_eq_deep(G, chi, psi))
{
flint_printf("FAIL: index\n\n");
flint_printf("q = %wu\n\n", q);
flint_printf("m = %wu\n\n", m);
flint_printf("chi = "); dirichlet_char_print(G, chi);
flint_printf("\n\nindex(chi) = %wu\n\n", p1);
flint_printf("psi(index) = %wu\n\n", psi->n);
flint_printf("psi = "); dirichlet_char_print(G, psi);
flint_printf("\n\n");
abort();
}
/* lift to higher modulus */
q2 = q * (1 + n_randint(state, 100));
dirichlet_group_init(G2, q2);
dirichlet_char_init(chi2, G2);
dirichlet_char_lift(chi2, G2, chi, G);
p1 = dirichlet_conductor_char(G, chi);
p2 = dirichlet_conductor_char(G2, chi2);
check_eq(p1, p2, q, m, "conductor chi", "conductor lift");
p1 = dirichlet_order_char(G, chi);
p2 = dirichlet_order_char(G2, chi2);
check_eq(p1, p2, q, m, "order chi", "order lift");
/* and lower */
dirichlet_char_lower(psi, G, chi2, G2);
if (!dirichlet_char_eq_deep(G, chi, psi))
{
flint_printf("FAIL: lift and lower back\n\n");
flint_printf("q = %wu\n\nchi = ", q);
dirichlet_char_print(G, chi);
flint_printf("\n\nq2 = %wu\n\nchi2 = ", q2);
dirichlet_char_print(G2, chi2);
flint_printf("\n\nq = %wu\n\npsi = ", q);
dirichlet_char_print(G, psi);
flint_printf("\n\n");
abort();
}
q3 = dirichlet_conductor_char(G, chi) * random_divisor(state, G);
q3 = n_gcd(q, q3);
dirichlet_group_init(G3, q3);
dirichlet_char_init(chi3, G3);
dirichlet_char_lower(chi3, G3, chi2, G2);
p1 = dirichlet_conductor_char(G, chi);
p2 = dirichlet_conductor_char(G3, chi3);
check_eq(p1, p2, q, m, "conductor chi", "conductor lower");
p1 = dirichlet_order_char(G, chi);
p2 = dirichlet_order_char(G3, chi3);
check_eq(p1, p2, q, m, "order chi", "order lower");
dirichlet_char_clear(chi3);
dirichlet_group_clear(G3);
dirichlet_char_clear(chi2);
dirichlet_group_clear(G2);
/* another random character */
do
n = n_randint(state, q);
while (n_gcd(q, n) > 1);
dirichlet_char_log(psi, G, n);
pairing = dirichlet_pairing(G, m, n);
cn = dirichlet_chi(G, chi, n);
cm = dirichlet_chi(G, psi, m);
if (pairing != cn || pairing != cm)
{
flint_printf("FAIL: pairing\n\n");
flint_printf("q = %wu\n\n", q);
flint_printf("m = %wu\n\n", m);
flint_printf("n = %wu\n\n", n);
flint_printf("chi(m,n) = %wu\n\n", pairing);
flint_printf("chi(m)(n) = %wu\n\n", cn);
flint_printf("chi(n)(m) = %wu\n\n", cm);
abort();
}
}
dirichlet_group_dlog_clear(G);
dirichlet_char_clear(chi);
dirichlet_char_clear(psi);
dirichlet_group_clear(G);
}
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}