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Properties

Label 1-1183-1183.1081-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.991 - 0.130i$
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  + (−0.903 − 0.428i)2-s + (−0.885 − 0.464i)3-s + (0.632 + 0.774i)4-s + (0.979 − 0.200i)5-s + (0.600 + 0.799i)6-s + (−0.239 − 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (0.822 + 0.568i)11-s + (−0.200 − 0.979i)12-s + (−0.960 − 0.278i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (−0.160 − 0.987i)18-s + i·19-s + (0.774 + 0.632i)20-s + ⋯
L(s)  = 1  + (−0.903 − 0.428i)2-s + (−0.885 − 0.464i)3-s + (0.632 + 0.774i)4-s + (0.979 − 0.200i)5-s + (0.600 + 0.799i)6-s + (−0.239 − 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (0.822 + 0.568i)11-s + (−0.200 − 0.979i)12-s + (−0.960 − 0.278i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (−0.160 − 0.987i)18-s + i·19-s + (0.774 + 0.632i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9544428463 - 0.06231879922i\)
\(L(\frac12)\) \(\approx\) \(0.9544428463 - 0.06231879922i\)
\(L(1)\) \(\approx\) \(0.7151214549 - 0.1368387784i\)
\(L(1)\) \(\approx\) \(0.7151214549 - 0.1368387784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.903 - 0.428i)T \)
3 \( 1 + (-0.885 - 0.464i)T \)
5 \( 1 + (0.979 - 0.200i)T \)
11 \( 1 + (0.822 + 0.568i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.996 + 0.0804i)T \)
31 \( 1 + (0.600 + 0.799i)T \)
37 \( 1 + (0.600 + 0.799i)T \)
41 \( 1 + (-0.534 - 0.845i)T \)
43 \( 1 + (0.919 - 0.391i)T \)
47 \( 1 + (0.160 - 0.987i)T \)
53 \( 1 + (0.692 + 0.721i)T \)
59 \( 1 + (-0.979 + 0.200i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (-0.935 + 0.354i)T \)
71 \( 1 + (-0.999 + 0.0402i)T \)
73 \( 1 + (-0.0804 + 0.996i)T \)
79 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (0.464 + 0.885i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.979 - 0.200i)T \)
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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

−21.22102535042625304982040895828, −20.57276012307252681707404617924, −19.444145853726704705867413375071, −18.75684708741061400699412407990, −17.903308976354632061084152983797, −17.33475469245066575776719082038, −16.76500678110915029151082492599, −16.209933683988308114879426412149, −15.0101485236736125092215335067, −14.68936636964428526690767558730, −13.51954609275842598112696688272, −12.53524286159991083933764701951, −11.370463679372187961148199858256, −10.92877729326377880060539265799, −10.12828460157629631034781392076, −9.35211125797503760174869833939, −8.83826333243006420022405045854, −7.557967248145732430200865026159, −6.45415895624823964365112579250, −6.189820200788406275319330428293, −5.338230934702633242075304697394, −4.30059422199179689763517578800, −2.90436576308125175637497779731, −1.68596718170518727115929986193, −0.72598595684194935408089902546, 1.06551970309798997651884947130, 1.63454836071956302063094514919, 2.620201603199606396611512746052, 3.95085090351727547884235849150, 5.173073723831816332683135404764, 5.99209753802435877571878293036, 6.91454121849964692192722230456, 7.44818600756878445544886810874, 8.65015150883259178343302094945, 9.52942315285461234034058006971, 10.08425791715906683531160260676, 10.912077413739980135767073186999, 11.87910879919672761050166987913, 12.27069060291821228970487028294, 13.221970516318065125347411212402, 13.96284455371695400477746861805, 15.203391772523128722222743540667, 16.29650773299044808122425197760, 16.88737576751110794316928615353, 17.37751268750082442669211377591, 18.122927959124275350809638805243, 18.67670793984766407966314723926, 19.50383197464561194399277194965, 20.48915345185085154611749640115, 21.04987439086155110456465141455

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