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Properties

Label 1-1183-1183.1166-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.258 - 0.965i$
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  + (0.120 − 0.992i)2-s + (−0.200 − 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)5-s + (−0.996 + 0.0804i)6-s + (−0.354 + 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (−0.919 − 0.391i)11-s + (−0.0402 + 0.999i)12-s + (0.987 − 0.160i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.278 + 0.960i)18-s + (−0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + (−0.200 − 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)5-s + (−0.996 + 0.0804i)6-s + (−0.354 + 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (−0.919 − 0.391i)11-s + (−0.0402 + 0.999i)12-s + (0.987 − 0.160i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.278 + 0.960i)18-s + (−0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5956321836 - 0.7761318754i\)
\(L(\frac12)\) \(\approx\) \(0.5956321836 - 0.7761318754i\)
\(L(1)\) \(\approx\) \(0.6732524978 - 0.4972355708i\)
\(L(1)\) \(\approx\) \(0.6732524978 - 0.4972355708i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.120 - 0.992i)T \)
3 \( 1 + (-0.200 - 0.979i)T \)
5 \( 1 + (-0.0402 + 0.999i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.919 + 0.391i)T \)
31 \( 1 + (0.428 - 0.903i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (0.948 + 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (0.278 - 0.960i)T \)
53 \( 1 + (0.987 - 0.160i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (-0.632 + 0.774i)T \)
67 \( 1 + (0.278 - 0.960i)T \)
71 \( 1 + (-0.200 - 0.979i)T \)
73 \( 1 + (0.799 - 0.600i)T \)
79 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.845 + 0.534i)T \)
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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

−21.23469081505796566249727938415, −21.02413633411744512372352284221, −20.13977919872775717619862824820, −19.03545643452123085375855300815, −18.03766002042858084419111228534, −17.297318835368593076037863012256, −16.69030141805398604597907452213, −15.93729893883148345155138698401, −15.56010740688289830206418631030, −14.67014855980204832550695790032, −13.844583662725816295846391458832, −12.90845078170457359051917181820, −12.31200088823763792890130565474, −11.162537854699069542741338712479, −10.1499605554521715190927603386, −9.37491952539178110473448795082, −8.78351687449071884395847787768, −7.95209250074024002458482634572, −7.04833324735133868178783087386, −5.74269119349522325415082790728, −5.32552097746927010258882175058, −4.49566816834016739957738715624, −3.877281909896166517378986552309, −2.57152874625051771902571083417, −0.71349482013942193887261078147, 0.62132478923087634700660927972, 1.96881980611789955163143555704, 2.61592470761607583145548953982, 3.37824756121148182332096949002, 4.612090712322693597684679500418, 5.69229282501095379471584090653, 6.39763427043224415180128885625, 7.458274478118281727997395017169, 8.21963757701229739525660426162, 9.13213362492189085933153740330, 10.29911488427961964699726383587, 11.04347914210886364505808437814, 11.3341218062529757676613762193, 12.44492587254639766301834427294, 13.31257614319011697080260530586, 13.47581634793826330574217945126, 14.74961819691799090567064696660, 15.138639052391271066778892468202, 16.68973751989504770821769435777, 17.55468546612592406429188815527, 18.14846157832247828925501969648, 18.83245499794513693702966760851, 19.30157567205410039338498474372, 20.0143419105633016707485393933, 21.0859655539679566662119084821

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