↑

Properties

Label 2-126-63.20-c3-0-7
Degree $2$
Conductor $126$
Sign $0.657 + 0.753i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−5.04 − 1.25i)3-s + (1.99 + 3.46i)4-s + (0.984 + 1.70i)5-s + (7.47 + 7.21i)6-s + (−17.4 + 6.07i)7-s − 7.99i·8-s + (23.8 + 12.6i)9-s − 3.93i·10-s + (15.6 + 9.06i)11-s + (−5.72 − 19.9i)12-s + (7.54 − 4.35i)13-s + (36.3 + 6.97i)14-s + (−2.81 − 9.83i)15-s + (−8 + 13.8i)16-s + 24.9·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.970 − 0.242i)3-s + (0.249 + 0.433i)4-s + (0.0880 + 0.152i)5-s + (0.508 + 0.491i)6-s + (−0.944 + 0.328i)7-s − 0.353i·8-s + (0.882 + 0.469i)9-s − 0.124i·10-s + (0.430 + 0.248i)11-s + (−0.137 − 0.480i)12-s + (0.161 − 0.0929i)13-s + (0.694 + 0.133i)14-s + (−0.0485 − 0.169i)15-s + (−0.125 + 0.216i)16-s + 0.356·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.705765 - 0.320612i\)
\(L(\frac12)\) \(\approx\) \(0.705765 - 0.320612i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 + (5.04 + 1.25i)T \)
7 \( 1 + (17.4 - 6.07i)T \)
good5 \( 1 + (-0.984 - 1.70i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15.6 - 9.06i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-7.54 + 4.35i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 24.9T + 4.91e3T^{2} \)
19 \( 1 + 147. iT - 6.85e3T^{2} \)
23 \( 1 + (-133. + 77.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-171. - 99.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (18.9 - 10.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 4.61T + 5.06e4T^{2} \)
41 \( 1 + (-209. - 363. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (161. - 279. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-102. + 176. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 512. iT - 1.48e5T^{2} \)
59 \( 1 + (-296. - 513. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-294. - 169. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (268. + 465. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 13.3iT - 3.57e5T^{2} \)
73 \( 1 - 27.3iT - 3.89e5T^{2} \)
79 \( 1 + (-230. + 399. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-43.2 + 74.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 336.T + 7.04e5T^{2} \)
97 \( 1 + (-155. - 89.6i)T + (4.56e5 + 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.60510679702133228816248844559, −11.64388347777562532397556394084, −10.70987782481467950148318142552, −9.772470618995101629555533885320, −8.700388493401676814149361105215, −7.02999101495129925068252649784, −6.40283915709087398099155876686, −4.77035338855002059684519524255, −2.81273763455171669723530530053, −0.75741439832996271712166512620, 1.02261962201238042103764637272, 3.73258928600709825043205874877, 5.46201038736323876747235059627, 6.39753601348233565945982928083, 7.43284881672964623931290649399, 9.020097927335630818903437966793, 9.942981214236579071553440543185, 10.75011499832251906330834325822, 11.91751870272027113026119633011, 12.84659724330180915055675928215

Graph of the $Z$-function along the critical line