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Properties

Label 1-189-189.160-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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)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 + (0.939 − 0.342i)13-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s − 26-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)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 + (0.939 − 0.342i)13-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)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: $\chi_{189} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6490319632 + 0.7293113174i\)
\(L(\frac12)\) \(\approx\) \(0.6490319632 + 0.7293113174i\)
\(L(1)\) \(\approx\) \(0.7108975190 + 0.1629954864i\)
\(L(1)\) \(\approx\) \(0.7108975190 + 0.1629954864i\)

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 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + T \)
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 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−26.62340407042272924494663427446, −25.79870503795853032054085823832, −24.70227789203985206072765409898, −24.08092411322647847704969209678, −23.2415116920993133168942854706, −21.54678945085680933536836961392, −20.71903588465250553645459199294, −19.72732195038238758332031560547, −18.93969098626026236913796922869, −17.88624117756814156790466639337, −16.714462980501033350452620504253, −16.29947570460608540907363110119, −15.21963836594184107784210170545, −13.95734040102903924399923456603, −12.73967125836444142127313969703, −11.46416409950350240362812325600, −10.64953185561819566118791650673, −9.08565375995882332078071421445, −8.72017848685481136635693482899, −7.53200335916358122350706977142, −6.19972430333118472470383510394, −5.20555035431714117624377998036, −3.51614885396517200214242323784, −1.622989065754646163221990180206, −0.50896836629643659259118844261, 1.373378571557264707387970270366, 2.79238104582134772684207565297, 3.831038619238178828994667687850, 5.87271902439251406991548864499, 7.16083824243125705661494168316, 7.78396312988984285333191397866, 9.27603667089523602009159802426, 10.11554675010577773347383736906, 11.14059145446704440310203470018, 11.93101505590056386993433807956, 13.222873930907180924532600188252, 14.6606395703229399762595310064, 15.551896548825341771998193940398, 16.59220284937035758000122156709, 17.81644593159423342029506951389, 18.40839951192874686479125971977, 19.26518545421533074106623885208, 20.400153445510411338203961155630, 21.087756515928206818872695508352, 22.4753747093707549411721538441, 23.07301305156828902175818414487, 24.64567527294596320640935985089, 25.63968364902101579548127066760, 26.10593675740209766801953775563, 27.40699598745036848910898249037

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