Properties

Label 1-189-189.97-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.116 + 0.993i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)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 + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)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 + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9449824331 + 1.061868170i\)
\(L(\frac12)\) \(\approx\) \(0.9449824331 + 1.061868170i\)
\(L(1)\) \(\approx\) \(0.8721767601 + 0.4544349927i\)
\(L(1)\) \(\approx\) \(0.8721767601 + 0.4544349927i\)

Euler product

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

−26.82036209318733011174304647115, −26.006920862499184405498906501878, −24.56211711986583116105918132713, −23.43343407977958346653871149459, −22.62094611099606153653761368386, −22.041371078352783722441670867124, −20.773057999879623892004646291376, −19.7535279687335170188326421032, −19.30644340732628783421731393018, −18.06390608019670029350108852106, −17.328738094982450260073761549968, −15.6015337617244265799735301667, −14.77406600366141582469459404779, −13.81202982158277897826462307282, −12.44913974880553457086891781382, −11.85885694025496969581051597760, −10.69328740888468425805811578972, −9.93295651647402350208836039233, −8.5725623002895346431808349515, −7.43894215795591371586824302354, −5.945539912031720610878746614340, −4.45841678771590888241331342073, −3.52832142697946085923882399755, −2.33528838904625731178008082791, −0.63750291062728615536334056400, 0.99959409974577457520193066130, 3.47827226333233086920371976973, 4.42529937636873165542124942534, 5.57485567280182399007325156792, 6.813151829615224983422830417501, 7.86585486115522842339942488589, 8.79682174405627676483373996951, 9.7438313533541892636524861731, 11.66816638778418994404866124993, 12.2730064409639222833539135518, 13.73206077872291086873418093519, 14.36544100434211776263934824357, 15.67611606531823561186922839805, 16.4003624161119732468472621687, 17.04326823133584449052720354814, 18.45578752248828116418220014783, 19.248493747265450875875215171, 20.47868711277352146823759796312, 21.645083617676435974523325783100, 22.60870938245876283183443529031, 23.56498417590376918887047905733, 24.3117129946777079899830150005, 25.011915015584654036670290970802, 26.17649354667686058995330424308, 27.17706525559917959263305095632

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