Properties

Label 1-189-189.40-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.960 - 0.276i$
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  + (0.766 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.960 - 0.276i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.960 - 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2151138515 + 1.523756852i\)
\(L(\frac12)\) \(\approx\) \(-0.2151138515 + 1.523756852i\)
\(L(1)\) \(\approx\) \(0.9186047891 + 0.8392125979i\)
\(L(1)\) \(\approx\) \(0.9186047891 + 0.8392125979i\)

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

−26.56244874035437686973085451952, −25.04881478975747703446605197406, −24.25056803326050518155488118572, −23.4519101731002381962337041915, −22.58038541333845380160656639721, −21.56268584918513627495824330381, −20.627085252965347166116896364763, −19.73652787423390386092559294591, −19.122361994499010509441000442589, −17.78626171955318802648320087933, −16.31141489075303747287898257107, −15.5822350843934216474892217624, −14.477630448410527974368993057304, −13.36055006810054354114532621345, −12.5627327800282138427673512915, −11.46451178054415762385771759041, −10.83474416117378050809101422719, −9.270661176874736896559923835418, −8.332886561973305238056679539, −6.667135693879080076727007508896, −5.57266956733997198628557584333, −4.22112645489628037936164848644, −3.55408679825145551648709437158, −1.80159029613047106162808404305, −0.401009234275179346985356020177, 2.242589114037340640875669492882, 3.79835849211268201281785198867, 4.37646068239254238318482953363, 6.14063702870413857967881299745, 6.84978808180992589636769312963, 7.96499443295731557038825805292, 9.040618612606101057931454813411, 10.83610919625273666946004163242, 11.677766531386530745189254042227, 12.69988940547370314240359703116, 13.84823356467421885691008303122, 14.82318919784054347393954492067, 15.49926691105722356750216689019, 16.499583463343280510817010008788, 17.5910546254066823400144487617, 18.64716444168372136457032055846, 19.89071212288314918565397313205, 20.808428262073492869090805864172, 22.1528652208671788307458787629, 22.592888615315752752345447760802, 23.641774576704855439840332293166, 24.30558701536605033547122641819, 25.64816211857022899774122393148, 26.11848921019334889012125013460, 27.25775836875535199841588895422

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