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Properties

Label 1-187-187.139-r0-0-0
Degree $1$
Conductor $187$
Sign $0.194 - 0.981i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
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  + (−0.891 − 0.453i)2-s + (−0.522 − 0.852i)3-s + (0.587 + 0.809i)4-s + (0.760 + 0.649i)5-s + (0.0784 + 0.996i)6-s + (−0.852 − 0.522i)7-s + (−0.156 − 0.987i)8-s + (−0.453 + 0.891i)9-s + (−0.382 − 0.923i)10-s + (0.382 − 0.923i)12-s + (0.951 − 0.309i)13-s + (0.522 + 0.852i)14-s + (0.156 − 0.987i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (−0.522 − 0.852i)3-s + (0.587 + 0.809i)4-s + (0.760 + 0.649i)5-s + (0.0784 + 0.996i)6-s + (−0.852 − 0.522i)7-s + (−0.156 − 0.987i)8-s + (−0.453 + 0.891i)9-s + (−0.382 − 0.923i)10-s + (0.382 − 0.923i)12-s + (0.951 − 0.309i)13-s + (0.522 + 0.852i)14-s + (0.156 − 0.987i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.194 - 0.981i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.194 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5225369527 - 0.4293187957i\)
\(L(\frac12)\) \(\approx\) \(0.5225369527 - 0.4293187957i\)
\(L(1)\) \(\approx\) \(0.6156776800 - 0.2753538567i\)
\(L(1)\) \(\approx\) \(0.6156776800 - 0.2753538567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (-0.522 - 0.852i)T \)
5 \( 1 + (0.760 + 0.649i)T \)
7 \( 1 + (-0.852 - 0.522i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.233 - 0.972i)T \)
31 \( 1 + (0.0784 - 0.996i)T \)
37 \( 1 + (-0.972 + 0.233i)T \)
41 \( 1 + (0.233 - 0.972i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.891 + 0.453i)T \)
59 \( 1 + (0.987 + 0.156i)T \)
61 \( 1 + (-0.996 + 0.0784i)T \)
67 \( 1 - T \)
71 \( 1 + (0.649 - 0.760i)T \)
73 \( 1 + (0.233 + 0.972i)T \)
79 \( 1 + (0.649 + 0.760i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.996 + 0.0784i)T \)
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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

−27.4570695033022494170424750266, −26.36777411302352685111490012531, −25.62527240570051038306919407424, −24.84314521986906736614663035274, −23.638413670376525036878076772611, −22.66966301198093911314390807112, −21.48578701287037309681469485501, −20.70003711817013721625483110446, −19.674297434199439181569913580439, −18.35774894020696240818829891668, −17.63420252517390244732157551324, −16.394716305957837678604167003666, −16.20002911750357194069674798175, −15.07461246765150555629886918204, −13.77617956300017982121583451194, −12.35637460665137914404746497753, −11.15605411414155604878038270076, −10.08169030161038423150647269718, −9.269078916513886818943229482161, −8.67290567629890490961756130912, −6.77986680125477613249520777532, −5.84116696934809450683934050713, −5.05828956806734702273115226728, −3.162615667423944314216529393385, −1.27464191508958335442554032536, 0.90278059657577617640383591248, 2.30774816619049057451264055994, 3.45995635887804520202082034161, 5.77471698811237971795280606680, 6.73470047956076098814257796348, 7.484814869790035031576403790518, 8.91569610990215479821784334262, 10.12864612821265361082209307145, 10.85098776982949476527564722382, 11.90040133215031477358298311530, 13.173340749774754746947606231386, 13.656166931707122185098023973051, 15.54092318150367003025768399755, 16.753526280456452924940513224191, 17.38543365851107485752410402469, 18.44960212416337843041111409456, 18.88485780518412180189885109881, 20.02704323988398709899801679360, 21.04653164551372402852312499454, 22.38747114400446265299228444426, 22.842556854276754560616964422021, 24.359478042251164713069250853326, 25.33522361136867288211636723241, 25.96729974967904523827757403043, 26.88310166479170397232674990747

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