Properties

Label 1-1183-1183.25-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.350 - 0.936i$
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  + (−0.948 + 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (−0.692 + 0.721i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (0.692 − 0.721i)12-s + (−0.568 + 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯
L(s)  = 1  + (−0.948 + 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (−0.692 + 0.721i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (0.692 − 0.721i)12-s + (−0.568 + 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6520437519 - 0.4522471468i\)
\(L(\frac12)\) \(\approx\) \(0.6520437519 - 0.4522471468i\)
\(L(1)\) \(\approx\) \(0.7640831615 + 0.002212430352i\)
\(L(1)\) \(\approx\) \(0.7640831615 + 0.002212430352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.948 + 0.316i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (-0.948 - 0.316i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.845 + 0.534i)T \)
37 \( 1 + (0.0402 - 0.999i)T \)
41 \( 1 + (0.354 - 0.935i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (-0.799 - 0.600i)T \)
53 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (-0.278 - 0.960i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (0.919 - 0.391i)T \)
71 \( 1 + (0.354 - 0.935i)T \)
73 \( 1 + (-0.948 - 0.316i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.970 - 0.239i)T \)
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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

−20.87416979711724126996574417209, −20.40276211512076573673541403293, −20.02783381101865059175097023783, −19.03354480678045247597790861139, −18.57050453957886315742659116241, −17.67983869265025292157661157092, −16.61096625786476817067951264343, −15.970719172830752816800682074749, −15.45583886325690619402061397588, −14.608641526432214312049714817943, −13.324842050912249960643853073424, −12.76358795474177534921530205919, −11.972205163319267070217830344928, −10.96337106570559482604376277643, −10.158023214404461952530227831, −9.38729885994770487760874105905, −8.63240126570144059167867728697, −7.93231811142762051499627562971, −7.52360197312944484523403791996, −6.33527409504442805117629042145, −4.83407243740943967698754836482, −4.02932050306532166151154991406, −3.05541447204586086801843659273, −2.19372152607848386642809073070, −1.18672184181359775420014326827, 0.39684093019911942603010940132, 1.95573027278087479708221235328, 2.692626708666362995719956595070, 3.51560155303982806212412544886, 4.76638423436124297081528082445, 6.075718664516925822901979650259, 7.002903823859925593249243700005, 7.62493764763541141338994404969, 8.18022348131699439361263550045, 9.09508835967252713794156925041, 9.80724320057486495755820590632, 10.79246579881092369600008785125, 11.31448551883426313051018898514, 12.41897317045539937266313916691, 13.55713116920620150649268393032, 14.18279104864680135704006979469, 15.22091081713997991757822156547, 15.61749380553446647079223963394, 16.1075039135549191559672330075, 17.58729199436017473972064964663, 18.07034474361695918049960617518, 18.83978378885185708650920314574, 19.48824150183022678577343062557, 19.9423696308263863835986288646, 20.82910239854383057268207092266

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