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Properties

Label 1-1183-1183.233-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.908 + 0.418i$
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  + (−0.278 − 0.960i)2-s + (0.799 + 0.600i)3-s + (−0.845 + 0.534i)4-s + (0.996 + 0.0804i)5-s + (0.354 − 0.935i)6-s + (0.748 + 0.663i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (−0.278 + 0.960i)11-s + (−0.996 − 0.0804i)12-s + (0.748 + 0.663i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.845 − 0.534i)18-s + (0.5 + 0.866i)19-s + (−0.885 + 0.464i)20-s + ⋯
L(s)  = 1  + (−0.278 − 0.960i)2-s + (0.799 + 0.600i)3-s + (−0.845 + 0.534i)4-s + (0.996 + 0.0804i)5-s + (0.354 − 0.935i)6-s + (0.748 + 0.663i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (−0.278 + 0.960i)11-s + (−0.996 − 0.0804i)12-s + (0.748 + 0.663i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.845 − 0.534i)18-s + (0.5 + 0.866i)19-s + (−0.885 + 0.464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.922305305 + 0.4218618126i\)
\(L(\frac12)\) \(\approx\) \(1.922305305 + 0.4218618126i\)
\(L(1)\) \(\approx\) \(1.344654975 - 0.04472879703i\)
\(L(1)\) \(\approx\) \(1.344654975 - 0.04472879703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.278 - 0.960i)T \)
3 \( 1 + (0.799 + 0.600i)T \)
5 \( 1 + (0.996 + 0.0804i)T \)
11 \( 1 + (-0.278 + 0.960i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (0.632 + 0.774i)T \)
37 \( 1 + (-0.987 + 0.160i)T \)
41 \( 1 + (-0.120 + 0.992i)T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (0.845 + 0.534i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (-0.428 - 0.903i)T \)
61 \( 1 + (0.948 + 0.316i)T \)
67 \( 1 + (0.0402 - 0.999i)T \)
71 \( 1 + (-0.120 + 0.992i)T \)
73 \( 1 + (-0.278 + 0.960i)T \)
79 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (-0.120 - 0.992i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.568 - 0.822i)T \)
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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

−21.23224790569712115294915970602, −20.31689920710259077845592153677, −19.32654524053285239800191542865, −18.76578536640960280111389055849, −18.06421282898244765028633778176, −17.38017356369322122082541342301, −16.62860104963841498394380963819, −15.70467715118099370143218787717, −14.90634675920459838892677160040, −14.095452871182176481612277802432, −13.55887875299891229436955585798, −13.11085495434568318038659628852, −11.961582224540446420701952032520, −10.616646570770334448084020574799, −9.7453778276242748863683805876, −9.11123255600834258205483786411, −8.396147398358339985485624698440, −7.56965448296275680055716307477, −6.828397389999844010586125127920, −5.81730207942234003662143713860, −5.41550088402063941313852254550, −3.952660925691133426583763145686, −2.96365038699057018861876518210, −1.77285354855818395692324437652, −0.837585159908356370401678715713, 1.41970339659796101455777149218, 2.17482780319307812274417590672, 2.96735904650981819693366279731, 3.85986665092777819296280884435, 4.85705092892536383540152454883, 5.55295059976691599142195673392, 7.09963242285452714645910995398, 8.00499629687466674885738863828, 8.8024267408894394884947190149, 9.739078092949719392452894991984, 10.03465023018363673238888254369, 10.666263381440334847418799122279, 11.934069479303393490745465906341, 12.67940689337619390512862480765, 13.51005317743580895173311509484, 14.21846256568285573095348273078, 14.73057878351159109958953771711, 15.97005344036169132832100753540, 16.8023676229598350014701901656, 17.53252985184369739388446204050, 18.53084490714175865076124871101, 18.84336283034657911256461062466, 20.09028357103802204945346385516, 20.55409649206534802847364715394, 21.02395933074775310690247433587

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