Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.194 - 0.981i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.194 - 0.981i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (139, \cdot )$
Sato-Tate  :  $\mu(80)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.194 - 0.981i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5225369527 - 0.4293187957i$
$L(\frac12,\chi)$  $\approx$  $0.5225369527 - 0.4293187957i$
$L(\chi,1)$  $\approx$  0.6156776800 - 0.2753538567i
$L(1,\chi)$  $\approx$  0.6156776800 - 0.2753538567i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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