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Properties

Label 1-189-189.4-r0-0-0
Degree $1$
Conductor $189$
Sign $0.983 - 0.178i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.983 - 0.178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9005942956 - 0.08115069351i\)
\(L(\frac12)\) \(\approx\) \(0.9005942956 - 0.08115069351i\)
\(L(1)\) \(\approx\) \(0.8470139080 + 0.005783245469i\)
\(L(1)\) \(\approx\) \(0.8470139080 + 0.005783245469i\)

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

−26.958534491942302234800752659163, −26.45599898874385493568160035770, −25.27060154811276592911425897982, −24.847140511059930702792653210793, −23.32671965133622781117304664390, −22.008888313225996825647071813360, −21.40606365591722299779229729418, −20.38415099492951171654057223823, −19.23716421178072650654678393054, −18.48820174354523388409017190249, −17.65345954791622938995000943079, −16.67970905081524214577761442064, −15.75922968929549757010812897077, −14.32638154879259995039521660511, −13.45316412107960874251505567010, −11.930691515954167873007466156505, −11.13442926480575975858611737946, −10.084644610254845751940987731141, −9.20316269018797242099171900473, −8.162043322159255469533453887413, −6.721940748175658163115065172161, −6.07855528177532526035080015545, −3.93231796730081783775012416990, −2.64733202438969851383983833425, −1.42399009256563541854181268214, 1.16703385731665824040211567662, 2.36497961863503588172927467442, 4.4324072582864212231870218899, 5.88458385745003305389278040809, 6.631785483860637087597700653131, 8.15346134589651125769814334104, 8.94469513552205620575732347193, 9.92173532803487896809773181471, 10.86847931592323185300440072757, 12.202540578912384469413948597316, 13.34331136192323573140520882022, 14.5927920117360820980080588163, 15.59615365849984061545417112729, 16.62232596406221327518820185649, 17.55794003529621215486055612231, 18.019174271931716387803930438293, 19.49737631134906734449881814526, 20.17680263451330459171178803285, 21.1209403867970572384889256268, 22.2947496166154523612157822803, 23.67557998825979147366811960851, 24.45956471407013200408001685222, 25.462860251881397277301944027819, 25.87454024614980630564537379471, 27.17436715134841579494551203074

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