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Properties

Label 1-1183-1183.643-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.778 - 0.627i$
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  + (0.960 − 0.278i)2-s + (0.919 − 0.391i)3-s + (0.845 − 0.534i)4-s + (0.822 + 0.568i)5-s + (0.774 − 0.632i)6-s + (0.663 − 0.748i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (−0.721 + 0.692i)11-s + (0.568 − 0.822i)12-s + (0.979 + 0.200i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.464 − 0.885i)18-s + (−0.866 − 0.5i)19-s + (0.999 + 0.0402i)20-s + ⋯
L(s)  = 1  + (0.960 − 0.278i)2-s + (0.919 − 0.391i)3-s + (0.845 − 0.534i)4-s + (0.822 + 0.568i)5-s + (0.774 − 0.632i)6-s + (0.663 − 0.748i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (−0.721 + 0.692i)11-s + (0.568 − 0.822i)12-s + (0.979 + 0.200i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.464 − 0.885i)18-s + (−0.866 − 0.5i)19-s + (0.999 + 0.0402i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.778 - 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.383059217 - 1.546775827i\)
\(L(\frac12)\) \(\approx\) \(4.383059217 - 1.546775827i\)
\(L(1)\) \(\approx\) \(2.735419210 - 0.6775224389i\)
\(L(1)\) \(\approx\) \(2.735419210 - 0.6775224389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.960 - 0.278i)T \)
3 \( 1 + (0.919 - 0.391i)T \)
5 \( 1 + (0.822 + 0.568i)T \)
11 \( 1 + (-0.721 + 0.692i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.278 + 0.960i)T \)
31 \( 1 + (-0.935 - 0.354i)T \)
37 \( 1 + (0.160 + 0.987i)T \)
41 \( 1 + (-0.391 - 0.919i)T \)
43 \( 1 + (-0.987 - 0.160i)T \)
47 \( 1 + (-0.464 - 0.885i)T \)
53 \( 1 + (-0.748 - 0.663i)T \)
59 \( 1 + (0.903 - 0.428i)T \)
61 \( 1 + (0.200 + 0.979i)T \)
67 \( 1 + (-0.534 + 0.845i)T \)
71 \( 1 + (0.600 - 0.799i)T \)
73 \( 1 + (0.239 + 0.970i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (-0.992 + 0.120i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.0804 - 0.996i)T \)
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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

−21.29585811066483471162398394128, −20.8711447055135602150769397507, −20.08660756781800478712502762123, −19.19361690795636856225859228642, −18.32126837871934896486947312464, −17.043031840041708536235820461760, −16.49394545429547929312023458712, −15.86653237945752092236815383479, −14.819532768450637650888568810148, −14.37554783576123746753939019798, −13.556376003097458155587687465658, −12.93942479494675407368029009804, −12.376590421370164126932008078299, −10.963282294231654470478315651545, −10.33640856123166430280752915984, −9.38494326432896762740358771880, −8.28948033793542386802579940593, −7.981380372513107132299336213752, −6.66114053802435282989805534533, −5.775118684475606904577779860521, −5.02512920629504854060905734693, −4.19263158992933013452752674941, −3.21461084171789324927560068776, −2.44099479710305401892652655745, −1.52230190663828515108541519955, 1.43023046858651946421142959983, 2.15611443470621747477276344501, 2.951393938169927278085790372940, 3.64311029226196242279163972624, 4.91254709934361372128143023813, 5.63371067483912335946673133262, 6.885891001582423995945445398697, 7.10736509913550055545832971444, 8.29414757251200053093305050793, 9.52909948658582896966372761439, 10.07065087890186321355544534414, 10.90525984210806340095159432341, 11.981429705922571674091124875791, 12.90090535830316610082245220857, 13.31064678855751316827133093332, 14.06720724718609813071885989144, 14.90369001954847250262318365594, 15.153200149026989065939905024829, 16.29307458410072160278713150776, 17.407757756115538211303566608115, 18.35667642571718198641353724084, 18.90483484581928988012624195996, 19.76760551247121639015170580144, 20.56277562807719155121876613912, 21.12913484914442266648840618517

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