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Properties

Label 2-130-65.24-c2-0-13
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

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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: $\chi_{130} (89, \cdot )$
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 \( 1 + (10.0 + 37.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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