void
acb_zeta(acb_t z, const acb_t s, slong prec)
{
acb_dirichlet_zeta(z, s, prec);
}
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);
}
}
