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Properties

Label 1-1183-1183.1172-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.733 + 0.679i$
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  + (−0.999 + 0.0402i)2-s + (0.970 + 0.239i)3-s + (0.996 − 0.0804i)4-s + (−0.160 + 0.987i)5-s + (−0.979 − 0.200i)6-s + (−0.992 + 0.120i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (0.464 + 0.885i)11-s + (0.987 + 0.160i)12-s + (−0.391 + 0.919i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (−0.903 − 0.428i)18-s + i·19-s + (−0.0804 + 0.996i)20-s + ⋯
L(s)  = 1  + (−0.999 + 0.0402i)2-s + (0.970 + 0.239i)3-s + (0.996 − 0.0804i)4-s + (−0.160 + 0.987i)5-s + (−0.979 − 0.200i)6-s + (−0.992 + 0.120i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (0.464 + 0.885i)11-s + (0.987 + 0.160i)12-s + (−0.391 + 0.919i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (−0.903 − 0.428i)18-s + i·19-s + (−0.0804 + 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4072694535 + 1.039311692i\)
\(L(\frac12)\) \(\approx\) \(0.4072694535 + 1.039311692i\)
\(L(1)\) \(\approx\) \(0.8032746719 + 0.4169737220i\)
\(L(1)\) \(\approx\) \(0.8032746719 + 0.4169737220i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.999 + 0.0402i)T \)
3 \( 1 + (0.970 + 0.239i)T \)
5 \( 1 + (-0.160 + 0.987i)T \)
11 \( 1 + (0.464 + 0.885i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.979 - 0.200i)T \)
37 \( 1 + (-0.979 - 0.200i)T \)
41 \( 1 + (0.721 + 0.692i)T \)
43 \( 1 + (-0.948 - 0.316i)T \)
47 \( 1 + (0.903 - 0.428i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (0.160 - 0.987i)T \)
61 \( 1 + (-0.120 + 0.992i)T \)
67 \( 1 + (-0.822 - 0.568i)T \)
71 \( 1 + (0.960 - 0.278i)T \)
73 \( 1 + (-0.534 + 0.845i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.239 - 0.970i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.160 + 0.987i)T \)
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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

−20.65149025915482354512393599943, −19.98453617977945866320161434864, −19.45911013678959344905205701622, −18.823921774141937588057877385366, −17.916026264441411077797637476683, −17.08007458849432911267408830332, −16.40758206832791566215044008346, −15.56567958226011372784057998162, −14.99834981200987283281895427558, −13.86459117029328659582365649436, −13.06656033106258243564479873597, −12.356631407455529323410096175965, −11.39111011960128960649489728045, −10.594738718161037400827771465822, −9.42242252171803561353967284104, −8.83162445313496558510151534454, −8.55498787184062974349032102634, −7.51961268125703436083023920789, −6.773288409768306271921122586345, −5.750741446854546081136042168093, −4.402156017828832847853808236162, −3.47495731265905888728380268414, −2.42631966233578167814570326945, −1.5383785960732988982798086797, −0.52724524962309797335776687701, 1.642831140635795287416402692226, 2.23896065574481326489428065232, 3.29037360522368189857188771171, 3.98904735158699780387224027367, 5.46156336902644630661263753398, 6.790680470462897584627128124340, 7.19682311533465809917451977244, 7.94758440034274596792973666018, 8.96187021845685108641548994400, 9.55649980584338026689195776279, 10.30278710565188728437537897196, 11.034434102467782956790715300349, 11.90053214464865935661953469377, 12.94269304105330815536319179291, 14.03662258283775535654702035716, 14.84640009044129800712954418108, 15.20817586565933382928801848064, 16.03516528998207421658851266883, 16.94795730449329905949157844693, 17.951558581262787176495268544676, 18.45210318557245998413205461544, 19.23172252108231057401602942148, 19.890201889382483339926593970364, 20.43653013882298253889785182776, 21.30893722981719634706707507489

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