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Properties

Label 1-189-189.47-r0-0-0
Degree $1$
Conductor $189$
Sign $0.508 + 0.860i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 + 0.642i)11-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.766 + 0.642i)5-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 + 0.642i)11-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.508 + 0.860i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.508 + 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842747285 + 1.051178349i\)
\(L(\frac12)\) \(\approx\) \(1.842747285 + 1.051178349i\)
\(L(1)\) \(\approx\) \(1.733298605 + 0.6264198939i\)
\(L(1)\) \(\approx\) \(1.733298605 + 0.6264198939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−26.9896561863288363073608541691, −25.80210415912732137680856318075, −24.72594059125416847517012886945, −24.14994006018378831294635069459, −23.20127102741072116081409969537, −21.91927114025935416951861412257, −21.39140807135946628042193962726, −20.538131378375672032791463904448, −19.51106053079323928345139105240, −18.47298210847517646356338316772, −17.03149005104049156956443436356, −16.243795909344964535246890556, −15.03424512214470363990838037001, −14.01264421645162440697381880290, −13.163978053169740630657906194011, −12.39785666842738561604766624391, −11.17048239433947123346089725311, −10.14149847864062571184549612240, −9.058956891798432072124831770018, −7.50867274151440221721391681867, −6.08039464149870516608304882527, −5.31468824000316126019830128564, −4.17046541269772294723681224677, −2.69251300429454282100764141920, −1.50970385768717269701588231044, 2.269194925227297625494129653269, 3.040101685676518260806358528961, 4.78019517668576781406599707798, 5.54821065965377075188409041876, 6.91569557779366318602696020688, 7.54132615492463180526045182490, 9.29615314507955471861713340396, 10.5212583448674113288521458789, 11.48553903142056677665766703074, 12.888617057815100753196905481727, 13.422408827037164666735520485487, 14.69960473683301497414092119173, 15.21906400347833650463533221174, 16.482678332380951094978301028473, 17.57585627880739039156394070717, 18.34902790433755665284470659582, 19.98063456778627647223280799363, 20.73726235402818351567157108688, 21.86909658963593401165401434020, 22.433464620187660579827457048672, 23.34690282767187067794842305537, 24.50478993440433429549235860367, 25.246515935230145115457261851651, 26.11152275460161488121472520838, 26.94032434523460642264523325383

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