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Properties

Label 1-57-57.2-r0-0-0
Degree $1$
Conductor $57$
Sign $0.363 + 0.931i$
Analytic cond. $0.264706$
Root an. cond. $0.264706$
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 + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)10-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 20-s + (−0.173 − 0.984i)22-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)10-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 20-s + (−0.173 − 0.984i)22-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(0.264706\)
Root analytic conductor: \(0.264706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (0:\ ),\ 0.363 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3889814964 + 0.2657064933i\)
\(L(\frac12)\) \(\approx\) \(0.3889814964 + 0.2657064933i\)
\(L(1)\) \(\approx\) \(0.5718398158 + 0.1363182903i\)
\(L(1)\) \(\approx\) \(0.5718398158 + 0.1363182903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
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

−32.61822169985772067961266707549, −32.062619917493803092351444392047, −30.09605971131694786400194055749, −29.2910222745877820331695056051, −27.8494213940177796161347201299, −27.23493672517042229720339089346, −26.11363882697384310422048269065, −24.84348903097687676487172786211, −23.87922812173791958995703915506, −22.79710263816355870653583447783, −20.778353224099032661614390142800, −19.79962133777482914692556018257, −19.03440767527460022845218385946, −17.393250000323694899019499610158, −16.46526831630748169263915003188, −15.58856275140263050797672232070, −13.99553982898558767333894238284, −12.27269892912724344726215633997, −10.93161325630149365446103027630, −9.64298096471959353386922397290, −8.26490881380570292275180108453, −7.26318636758794309652146291224, −5.597270251212545215093152269816, −3.51969864011884770944816386629, −0.84656680130504739234113178007, 2.247606403909790497838463661827, 3.81774766261026492683338722034, 6.41034421827540607893507371303, 7.59242605199769852746969867797, 9.02892623866699605129810739892, 10.19054644637308718461141400585, 11.69748802762087013683641327900, 12.37692540697505451219998705495, 14.67229535971909830846665293114, 15.75881686887092209128198857930, 16.94209484038354310838340659427, 18.48227500065303104932219329280, 19.07244340215403626307581270721, 20.20157995885737346010699870077, 21.65369542650996875044465350002, 22.70364927065016392043198144120, 24.30493511699158238598094196125, 25.65935084870748473699022682791, 26.35947096617376832405164581434, 27.735264021229073673382553531930, 28.3398215382151811140008948273, 29.74345110442553675367891486763, 30.7244446523473486792007958148, 31.727330779972979350180262784941, 33.546512068600299917678286493958

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