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Properties

Label 1-1183-1183.326-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.673 + 0.739i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.632 − 0.774i)2-s + (0.428 − 0.903i)3-s + (−0.200 + 0.979i)4-s + (0.799 + 0.600i)5-s + (−0.970 + 0.239i)6-s + (0.885 − 0.464i)8-s + (−0.632 − 0.774i)9-s + (−0.0402 − 0.999i)10-s + (−0.632 + 0.774i)11-s + (0.799 + 0.600i)12-s + (0.885 − 0.464i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.200 + 0.979i)18-s + (−0.5 + 0.866i)19-s + (−0.748 + 0.663i)20-s + ⋯
L(s)  = 1  + (−0.632 − 0.774i)2-s + (0.428 − 0.903i)3-s + (−0.200 + 0.979i)4-s + (0.799 + 0.600i)5-s + (−0.970 + 0.239i)6-s + (0.885 − 0.464i)8-s + (−0.632 − 0.774i)9-s + (−0.0402 − 0.999i)10-s + (−0.632 + 0.774i)11-s + (0.799 + 0.600i)12-s + (0.885 − 0.464i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.200 + 0.979i)18-s + (−0.5 + 0.866i)19-s + (−0.748 + 0.663i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6753334676 + 0.2983550030i\)
\(L(\frac12)\) \(\approx\) \(0.6753334676 + 0.2983550030i\)
\(L(1)\) \(\approx\) \(0.7660582859 - 0.2250801716i\)
\(L(1)\) \(\approx\) \(0.7660582859 - 0.2250801716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.632 - 0.774i)T \)
3 \( 1 + (0.428 - 0.903i)T \)
5 \( 1 + (0.799 + 0.600i)T \)
11 \( 1 + (-0.632 + 0.774i)T \)
17 \( 1 + (-0.845 - 0.534i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.354 + 0.935i)T \)
31 \( 1 + (0.692 + 0.721i)T \)
37 \( 1 + (0.278 - 0.960i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (-0.200 - 0.979i)T \)
53 \( 1 + (-0.845 - 0.534i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (-0.845 + 0.534i)T \)
67 \( 1 + (0.948 + 0.316i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (-0.632 + 0.774i)T \)
79 \( 1 + (-0.200 - 0.979i)T \)
83 \( 1 + (0.568 - 0.822i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.120 + 0.992i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.01953660331820485021940228234, −20.36207393504143033912867158556, −19.572853316460405217911404887547, −18.78038536684401622606352098683, −17.84220849770362181731764095420, −16.99680902792103215125112550594, −16.66047942460877954338365827417, −15.572238659124132867150011088506, −15.343223241748536306583046554142, −14.14097500113797285730148687746, −13.654135988242748929103296869520, −12.87879682865060189752889018467, −11.237781952558822706106004448105, −10.63291154642950115084675462701, −9.81931772466198000903900167441, −9.18032663899596014964592541640, −8.41362254589898524286675421343, −7.9503609439751422591349940633, −6.40380354623924991994764398440, −5.90361469345465450319559622922, −4.83661303873819987319038347159, −4.3539877694422612423997451562, −2.74903019503083641369994632111, −1.89671199770457337359554101382, −0.32306265777792686520429635233, 1.430125098250504528369254933750, 2.087752394496284687311127099874, 2.79183344769570686396764483998, 3.720878081687036472818597786714, 5.075025208479547345314917548811, 6.30484165941048235007035906770, 7.107267960743815380216226335, 7.7616951200454921599129285037, 8.70404851069402068707627801371, 9.5082735873259294947283167520, 10.20859554623092005915504248990, 11.061299908286196943377097411631, 11.94328147918281367745142345193, 12.78765903189964288810976504690, 13.328744489997956058449868902176, 14.09692690841971122769229343545, 14.939758953057455966537812258626, 16.05752476429543409409811253823, 17.18783718737879452894535897916, 17.834572381506699330784650444124, 18.24300924832114983336199284630, 18.88639099319962876430285817928, 19.83162731889270744339436393628, 20.324319513480202674017608398780, 21.212326248064574260843361790562

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