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Properties

Label 1-1183-1183.417-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: $\chi_{1183} (417, \cdot )$
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

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

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