↑

Properties

Label 1-189-189.122-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

Related objects

Downloads

Learn more

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: $\chi_{189} (122, \cdot )$
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 \( 1 + (-0.173 + 0.984i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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