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Properties

Label 2-130-65.58-c1-0-2
Degree $2$
Conductor $130$
Sign $0.992 + 0.123i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (1.18 − 0.316i)3-s + (0.499 + 0.866i)4-s + (1.44 + 1.70i)5-s + (−1.18 − 0.316i)6-s + (0.401 + 0.695i)7-s − 0.999i·8-s + (−1.30 + 0.752i)9-s + (−0.401 − 2.19i)10-s + (3.73 − 1.00i)11-s + (0.864 + 0.864i)12-s + (−2.61 − 2.48i)13-s − 0.803i·14-s + (2.24 + 1.55i)15-s + (−0.5 + 0.866i)16-s + (1.73 − 6.46i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.682 − 0.182i)3-s + (0.249 + 0.433i)4-s + (0.647 + 0.762i)5-s + (−0.482 − 0.129i)6-s + (0.151 + 0.262i)7-s − 0.353i·8-s + (−0.434 + 0.250i)9-s + (−0.127 − 0.695i)10-s + (1.12 − 0.302i)11-s + (0.249 + 0.249i)12-s + (−0.725 − 0.688i)13-s − 0.214i·14-s + (0.580 + 0.401i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07064 - 0.0664190i\)
\(L(\frac12)\) \(\approx\) \(1.07064 - 0.0664190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.44 - 1.70i)T \)
13 \( 1 + (2.61 + 2.48i)T \)
good3 \( 1 + (-1.18 + 0.316i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.401 - 0.695i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.73 + 1.00i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.73 + 6.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.33 - 4.99i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.912 + 3.40i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.81 + 3.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.38 - 3.38i)T - 31iT^{2} \)
37 \( 1 + (0.152 - 0.263i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.821 - 3.06i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.55 + 0.684i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 + (-1.89 - 1.89i)T + 53iT^{2} \)
59 \( 1 + (-10.8 - 2.91i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.81 + 3.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.521 - 0.301i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.02 - 1.61i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 8.89iT - 73T^{2} \)
79 \( 1 - 7.04iT - 79T^{2} \)
83 \( 1 - 3.58T + 83T^{2} \)
89 \( 1 + (4.40 + 16.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.40 - 3.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.41046527322830646241844300411, −12.09069597895333639697561594750, −11.18222203077898509119864406457, −9.997306698037618146151757313536, −9.211968821670681496771767909000, −8.083118720259531663968390133370, −7.03011269833288792082446929125, −5.61193283520694817613317643777, −3.34411594510023973552728246087, −2.17176010838433026691600888337, 1.85399672613982573458467893935, 4.04647818674869499568921085872, 5.65018811693081735842496319722, 6.92044492020403082143447327663, 8.294057270461117983414840312297, 9.189001858883392368190997549578, 9.686652317451950898471520018973, 11.15885983676303558872226128369, 12.34686887380566576954585195347, 13.53210834101717649488994625563

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