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Properties

Label 1-133-133.89-r0-0-0
Degree $1$
Conductor $133$
Sign $0.0471 - 0.998i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
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  + (0.939 − 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0471 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0471 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.0471 - 0.998i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.0471 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8596845458 - 0.8200934013i\)
\(L(\frac12)\) \(\approx\) \(0.8596845458 - 0.8200934013i\)
\(L(1)\) \(\approx\) \(1.091272439 - 0.4800963594i\)
\(L(1)\) \(\approx\) \(1.091272439 - 0.4800963594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
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

−28.905147195603106874056661899169, −28.15562967340503676368261340775, −26.691983678403546005047982452142, −25.80972256302518040724459958160, −24.47972379416881994093583791610, −23.610098213012788272948852793059, −23.00012481411410314098245186032, −22.22775454172403087124865860144, −21.15937317566396054591841990580, −19.861243158386303530723824081823, −18.59448153237558178380517409773, −17.590288207661178939793324084133, −16.329780610481449832073591765707, −15.58449967373747149150716977217, −14.52639691830861209412533014198, −13.163822989365100182294736683029, −12.30593016425727186952812220773, −11.28839470750849183434164061517, −10.54118259111934304175929152819, −8.21046695318101795570328769629, −7.01147931161619661224701345710, −6.38833767523605191836145588944, −4.902398946623952109416586580125, −3.912770667134363010885890308996, −2.17140949023112407621220000491, 0.947026418645129920819922290494, 3.23011873279413198561243598617, 4.41155140327230945841740515523, 5.362109589154158877427926547442, 6.42177681769904329165244779947, 7.97698578225012513177721694062, 9.700244676277805577316016376946, 11.13651813137160466902483260964, 11.48914039577206795664291573486, 12.77370037951894008576673203209, 13.55896610065571767811966033981, 15.431147456776692041278033304546, 15.76185594319929033400954611634, 16.829654262869805440890692226661, 18.31628638802154719785294373222, 19.51232472557533053594500393551, 20.65979475858384008724372668448, 21.36005818775031103410134098803, 22.54659708271161009924804132150, 23.26524571896933480214229171780, 24.034489197910413009433701611882, 24.918880428245399089666914213338, 26.68760751614887979218578622123, 27.715726361387673932950597348115, 28.49165952088890031825417757153

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