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Properties

Label 1-189-189.121-r0-0-0
Degree $1$
Conductor $189$
Sign $0.545 + 0.838i$
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 + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + 10-s + (0.766 + 0.642i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s + 17-s + 19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + 10-s + (0.766 + 0.642i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s + 17-s + 19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684522775 + 0.9139449847i\)
\(L(\frac12)\) \(\approx\) \(1.684522775 + 0.9139449847i\)
\(L(1)\) \(\approx\) \(1.570577210 + 0.5910230526i\)
\(L(1)\) \(\approx\) \(1.570577210 + 0.5910230526i\)

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

−27.084289362395617805177218669726, −26.01195921429748562530278030185, −24.77896327416030831895355006632, −24.17361930528095755482000829842, −22.82044033262753428860014226006, −22.12168795023795266282096068784, −21.47244105392368730783090578880, −20.43205467566517366226742847613, −19.30811240522119907317577511708, −18.60524757015851619839999807773, −17.383184088332911459287930998124, −16.18207162594141978243806053481, −14.72816481109999461253040379561, −14.22138305394219732881767113197, −13.322621333595743906485926476410, −12.02678124174112003532438881351, −11.27080609411698364096231765733, −9.97951536419003007018070126593, −9.42265976204002803125464696695, −7.43030704907919391985738897788, −6.18698353855869258668686850451, −5.385767102778221797585692680815, −3.86343521907362467718261913640, −2.77974489525715623672161556856, −1.526879207148818731763326419889, 1.84704416649776700366415900418, 3.38590370701298534874872058178, 4.79827244293101398874804024280, 5.5452734426576343839512044388, 6.77299449560103740542578545187, 7.851497450143045019000984162745, 9.12158715666793622787641130569, 10.13345631875588707825486479563, 12.03903158527541083815344983554, 12.41702551214979229886315726448, 13.74922802270636586697981562626, 14.39387285673949719754988782915, 15.5086157189298583474995624892, 16.7090038662211470722013645869, 17.21922019740964726926713947546, 18.27915874163396126963709956028, 20.04099170541004998211776653734, 20.6377066531345436819297639576, 21.88333167559893381952487335118, 22.3632177028071958193389939206, 23.60285064664992326082036817566, 24.55504165432280665438752894765, 25.13954997129887916663211184141, 25.980621540350973656206791151352, 27.15487946387516100314719836224

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