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Properties

Label 1-189-189.20-r0-0-0
Degree $1$
Conductor $189$
Sign $-0.597 + 0.802i$
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 + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)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 + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + 26-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)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 + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ -0.597 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2291753125 + 0.4563255653i\)
\(L(\frac12)\) \(\approx\) \(0.2291753125 + 0.4563255653i\)
\(L(1)\) \(\approx\) \(0.5260074133 + 0.2746838062i\)
\(L(1)\) \(\approx\) \(0.5260074133 + 0.2746838062i\)

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

−26.79788062899531745352340531208, −26.36626530049804518484489276966, −24.76490197073545984417382276288, −24.27716547808574835660633997535, −22.68527659069632802187687736766, −22.00815831368833604383822295845, −20.74027792816397214676976680785, −19.90655532788921794916989856701, −19.26074514682392738698591609898, −18.30058739269270096227032346205, −17.04155662508785966001562927311, −16.3969709188592959393257236348, −15.3097214102425573139096326117, −13.89173122880474446894165933868, −12.58536736105172431852242014132, −11.689826033325664614065820886841, −11.107236032207419241453456828549, −9.52182538792016167040246271085, −8.879179049155594185618035753765, −7.65310716198190666779967015772, −6.76965834004754493419093320271, −4.67640120545628721324544891414, −3.68797261592474572596718355810, −2.27187383692820402007915439646, −0.53579303294997897073792924055, 1.56459714416068312455059166094, 3.45740617549636388671578960969, 4.889319286099491510175842985684, 6.287919507880016251502701965003, 7.328403644337128228356653702291, 8.11392382793946314948796848190, 9.31036825676618052782972555672, 10.38269193947215748098527143006, 11.41992082955568274817990101455, 12.4903732024210602426253294481, 14.21421830116763323886960861949, 14.96759584313148271734748265651, 15.76192055306726253461541522906, 16.86656706183381822998466491710, 17.70680661449240465747764498778, 18.788247084694601236940501612866, 19.70249132643745065701816903088, 20.19899707947674181840626664430, 22.02928434694445193233020605524, 22.89075453358097575504671779127, 23.8294982099186447967696498076, 24.69910353038939930648288594600, 25.61133276006274890971930868137, 26.65436305255821141124457838486, 27.37701419543421772394438884518

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