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Properties

Label 1-189-189.59-r0-0-0
Degree $1$
Conductor $189$
Sign $0.490 + 0.871i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)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 + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)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 + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.490 + 0.871i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.490 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5925859211 + 0.3462487061i\)
\(L(\frac12)\) \(\approx\) \(0.5925859211 + 0.3462487061i\)
\(L(1)\) \(\approx\) \(0.7081269386 + 0.07992053504i\)
\(L(1)\) \(\approx\) \(0.7081269386 + 0.07992053504i\)

Euler product

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

−26.85551993633735696623429415288, −26.14935560915976843331973197065, −24.94247655562223900950053060470, −24.26788387137284021254279353162, −23.68769294217498145635311751707, −22.24134643169335839263959282921, −21.04401596900599792633939282230, −20.07341472578763921434285220182, −19.1849350010354226669016324707, −18.15006011899561046604669296236, −17.180847584461595758276482011173, −16.25003836867843232729221924383, −15.756103351120371669623106452576, −14.1741559892971535171566010385, −13.50816408995621712696504187744, −11.9406259823837045896012135910, −10.92592180787891813865358142798, −9.501616225588817649267600630386, −8.91593878066542353233980425808, −7.86098937597596553521665258533, −6.60884680672832257355792381816, −5.49278453478733779421432255159, −4.43831982546875312310113420326, −2.26966804972312304482673448270, −0.697594061189452859001313377488, 1.754704513425964360998335325, 2.88579227453924662752734317377, 4.06389190818392793017235630134, 5.93709331056630743812250683389, 7.29461181498846211164580981224, 8.01711982183463288992819615168, 9.5817788630365743297879660724, 10.26186638341822213957001089275, 11.16258099614531859237062446671, 12.32402067919987137047476668377, 13.267055624279958870119739126896, 14.6987297604593132912768581109, 15.60653027680611061957273406594, 16.96405701206274567295087430736, 17.956801905501484792482913572088, 18.39357547775193712410417322357, 19.693815646950038425684552567531, 20.31193566225635847489899193651, 21.56710821488669911893118406685, 22.30866408870582156426817376990, 23.20537820803043723127875426084, 24.81236363563703924690600304936, 25.72331353295165534226327263005, 26.33386186130783415908912686447, 27.37475032869738554524197881665

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