↑

Properties

Label 2-130-65.19-c2-0-6
Degree $2$
Conductor $130$
Sign $0.883 - 0.468i$
Analytic cond. $3.54224$
Root an. cond. $1.88208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (3.51 + 2.02i)3-s + (1.73 − i)4-s + (−3.38 + 3.67i)5-s + (5.53 + 1.48i)6-s + (0.953 + 0.255i)7-s + (1.99 − 2i)8-s + (3.72 + 6.44i)9-s + (−3.28 + 6.26i)10-s + (11.6 − 3.13i)11-s + 8.10·12-s + (−12.9 − 0.497i)13-s + 1.39·14-s + (−19.3 + 6.04i)15-s + (1.99 − 3.46i)16-s + (−12.3 − 21.3i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (1.17 + 0.675i)3-s + (0.433 − 0.250i)4-s + (−0.677 + 0.735i)5-s + (0.923 + 0.247i)6-s + (0.136 + 0.0364i)7-s + (0.249 − 0.250i)8-s + (0.413 + 0.716i)9-s + (−0.328 + 0.626i)10-s + (1.06 − 0.284i)11-s + 0.675·12-s + (−0.999 − 0.0382i)13-s + 0.0996·14-s + (−1.29 + 0.402i)15-s + (0.124 − 0.216i)16-s + (−0.724 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.52139 + 0.627291i\)
\(L(\frac12)\) \(\approx\) \(2.52139 + 0.627291i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (3.38 - 3.67i)T \)
13 \( 1 + (12.9 + 0.497i)T \)
good3 \( 1 + (-3.51 - 2.02i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-0.953 - 0.255i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-11.6 + 3.13i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (12.3 + 21.3i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-8.03 - 2.15i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (0.823 - 1.42i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (22.0 - 38.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-28.0 + 28.0i)T - 961iT^{2} \)
37 \( 1 + (-2.55 - 9.53i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-5.33 - 19.9i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (17.8 + 30.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (50.2 - 50.2i)T - 2.20e3iT^{2} \)
53 \( 1 + 21.0iT - 2.80e3T^{2} \)
59 \( 1 + (21.7 - 81.3i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-28.7 - 49.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (41.5 - 11.1i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-92.5 - 24.7i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-31.7 + 31.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 13.5T + 6.24e3T^{2} \)
83 \( 1 + (-37.1 - 37.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-51.4 + 13.7i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (10.0 - 37.6i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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

−13.49145507166170719640393228558, −11.97676071544591715812537872014, −11.29865060127418274484673174959, −9.970716420068271861137703406302, −9.082109937734170491497123912587, −7.74735706073984124991427918268, −6.67103047683553076176423402205, −4.74714935208269965785604556357, −3.63668054849380571380094451389, −2.64996803728959211566304253685, 1.88560923654692852329151029494, 3.59156451522935849744733290950, 4.75145115659642299296251903769, 6.58234332165265805168878918116, 7.68872376953693594701700889550, 8.450240168497222485393952257420, 9.541831675116246651769626186690, 11.37126045657864839709439792292, 12.34101832836743393872880964397, 13.00950963480862496287427557834

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