Properties

Label 1-1183-1183.261-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.949 + 0.315i$
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  + (−0.996 − 0.0804i)2-s + (−0.0402 + 0.999i)3-s + (0.987 + 0.160i)4-s + (−0.200 + 0.979i)5-s + (0.120 − 0.992i)6-s + (−0.970 − 0.239i)8-s + (−0.996 − 0.0804i)9-s + (0.278 − 0.960i)10-s + (−0.996 + 0.0804i)11-s + (−0.200 + 0.979i)12-s + (−0.970 − 0.239i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (0.987 + 0.160i)18-s + (−0.5 − 0.866i)19-s + (−0.354 + 0.935i)20-s + ⋯
L(s)  = 1  + (−0.996 − 0.0804i)2-s + (−0.0402 + 0.999i)3-s + (0.987 + 0.160i)4-s + (−0.200 + 0.979i)5-s + (0.120 − 0.992i)6-s + (−0.970 − 0.239i)8-s + (−0.996 − 0.0804i)9-s + (0.278 − 0.960i)10-s + (−0.996 + 0.0804i)11-s + (−0.200 + 0.979i)12-s + (−0.970 − 0.239i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (0.987 + 0.160i)18-s + (−0.5 − 0.866i)19-s + (−0.354 + 0.935i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.949 + 0.315i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.949 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6250072608 + 0.1010324045i\)
\(L(\frac12)\) \(\approx\) \(0.6250072608 + 0.1010324045i\)
\(L(1)\) \(\approx\) \(0.5638687115 + 0.1894414170i\)
\(L(1)\) \(\approx\) \(0.5638687115 + 0.1894414170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.996 - 0.0804i)T \)
3 \( 1 + (-0.0402 + 0.999i)T \)
5 \( 1 + (-0.200 + 0.979i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
17 \( 1 + (0.692 - 0.721i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (0.799 + 0.600i)T \)
37 \( 1 + (-0.919 + 0.391i)T \)
41 \( 1 + (0.885 - 0.464i)T \)
43 \( 1 + (0.120 + 0.992i)T \)
47 \( 1 + (0.987 - 0.160i)T \)
53 \( 1 + (0.692 - 0.721i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (0.692 + 0.721i)T \)
67 \( 1 + (-0.632 - 0.774i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.996 + 0.0804i)T \)
79 \( 1 + (0.987 - 0.160i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.748 + 0.663i)T \)
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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

−20.80259709960954471231838237023, −20.330687960191653816555008454699, −19.215761664039881308775594179615, −19.131350404846036236314926729785, −18.03661239205114105682352376727, −17.44818176114255607486030421779, −16.73093984259252782028799472785, −16.04508022711935421008971942472, −15.20460986784031018302036596900, −14.15543479721940062249705690444, −13.193516609609596060821647287256, −12.34303887784394065417071255889, −11.99462117283280271068684744534, −10.86531324677886012469819415305, −10.10742560794917484160955351969, −9.046189947271287379003699344301, −8.206380045521789085823693606180, −7.9083922390481714790173317299, −6.99795893185277459357729809070, −5.82603040254453167987010686797, −5.44294491238099308036276048236, −3.81517074380086966875781465648, −2.57885023511694144779739345984, −1.67517048766015728413863375921, −0.84135875043892682579136298924, 0.474680906099874568279301146772, 2.47306195459495877654766749941, 2.7788377756004810470446461023, 3.890347156368024683177196118276, 5.046711080928134467312132694861, 6.09000369532164016510366681109, 6.92330211350045690386890121486, 7.88168409921176282556358224245, 8.55782748096804995688190653169, 9.59724225632409561942160562078, 10.293671996614948402074424928604, 10.69066496840961296853529541840, 11.54312532585045467947524534228, 12.25696563751635431943790564266, 13.69395904824250621240603787336, 14.58773687350442023003030669917, 15.35980509152950970205624244299, 15.8602231452696129555327479236, 16.54541865228928914368136466020, 17.636170561066656875767944955172, 18.009812891109698871417576037977, 19.06154102225808000790002132151, 19.52131946154064191502815619061, 20.63492050409669124831467163157, 21.042980935176101966122080769880

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