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Properties

Label 1-189-189.110-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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)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 + (0.939 − 0.342i)13-s + (0.766 − 0.642i)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 + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)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 + (0.939 − 0.342i)13-s + (0.766 − 0.642i)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 + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)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: $\chi_{189} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.490 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7540211320 - 0.4405755047i\)
\(L(\frac12)\) \(\approx\) \(0.7540211320 - 0.4405755047i\)
\(L(1)\) \(\approx\) \(0.7824716846 - 0.3303067654i\)
\(L(1)\) \(\approx\) \(0.7824716846 - 0.3303067654i\)

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

−27.05675011483237664009979834343, −26.48947807691300356822546807516, −25.23291243498406501134497975595, −24.4753926275468533543826570854, −23.59986043489516852346051968035, −22.849101384812427221577964897435, −21.84633998858922436662670307245, −20.431795421506900585870471436335, −19.33701620727260539467951142159, −18.639194906441392291896509688745, −17.38295497484866636884840302172, −16.44418198525002230564601007011, −15.7778742370182472535945541478, −14.72611900817581551710635998760, −13.79874373845554364118130149317, −12.583637855429455412890794548032, −11.46981212630747884005220062679, −10.12963012284951385793374928317, −8.67134062019514647228998988711, −8.295248341623469072156393878615, −6.87622303655604087613114205913, −5.993675943195575350520000706098, −4.4705062509534681025193555890, −3.699057228512303249379649020386, −1.12740538936011007351281908263, 1.04690221748761119852125313469, 2.8002044856187793652853440693, 3.81107292014652830005149442655, 4.858300070579729368966146002842, 6.696840032206592807887943201421, 7.94954968946869485431720128464, 8.97812595495049702264398316230, 10.05884279865352409578418332821, 11.49864389119413054316497145211, 11.57657729489216970704032373364, 13.04276953920131698882906549706, 14.001996151328538621105134389616, 15.22301487815972061451248426343, 16.28948370832149883960305123773, 17.69450161434457710467564060827, 18.34458209218349817130581240631, 19.636128104091561587731991707256, 19.925684177059595315260389503241, 21.12064286186150966954709557139, 22.28563841816261221189978406692, 22.87430757544779974971675038004, 23.821225781739408197598064486990, 25.242404217142696421638740464140, 26.33863614741856747304649066313, 27.22661379909365039776694594628

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