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Properties

Label 1-1027-1027.181-r0-0-0
Degree $1$
Conductor $1027$
Sign $0.412 + 0.911i$
Analytic cond. $4.76936$
Root an. cond. $4.76936$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + 10-s + (0.5 + 0.866i)11-s + 12-s + 14-s − 15-s + (−0.5 + 0.866i)16-s + 17-s − 18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + 10-s + (0.5 + 0.866i)11-s + 12-s + 14-s − 15-s + (−0.5 + 0.866i)16-s + 17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.412 + 0.911i$
Analytic conductor: \(4.76936\)
Root analytic conductor: \(4.76936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (0:\ ),\ 0.412 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387724929 + 0.8954484064i\)
\(L(\frac12)\) \(\approx\) \(1.387724929 + 0.8954484064i\)
\(L(1)\) \(\approx\) \(1.233279856 + 0.1555583165i\)
\(L(1)\) \(\approx\) \(1.233279856 + 0.1555583165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−21.783457992606523262203773057845, −20.71823543824331804089805439746, −20.00819433383647873727993640958, −18.90380128395633260442696760990, −17.99496586017696714514021886980, −17.339894332990495650134034209261, −16.70096673175306418278776688296, −16.38373681224323917479649423213, −15.08429164120460515846793459432, −13.893918458331087759204042014518, −13.71352395036234232109014643190, −13.02291505174647235283448794354, −11.93832720814741746724864702147, −11.5232712216589213551114638858, −10.17337739266597952971959021681, −9.02443229490898637194331989766, −8.21028511839394864331195318772, −7.515147167800431409149563240162, −6.69611269177802029729779559164, −5.72348956506333640681780842815, −5.246633134970729223292659808499, −4.26188423621309693183037117357, −3.13918151458372837110899145600, −1.596572299543098093815180703729, −0.672348359216208805783653811476, 1.442161548210512736531056414902, 2.42884484417360605236720112409, 3.34352923232720545949659744808, 4.2332409022412848377312922295, 5.20425036410969468788099945813, 5.8373261380017545016865209961, 6.6182367143398747874266814857, 8.14827835609067883014121242075, 9.374841270271296463507683537084, 9.86269151135176189266985514503, 10.503392116116850998648856217226, 11.46828191434623862702618265686, 11.96540620680784435257731802505, 12.71855083841441890532535817231, 14.18381487602375836668081228372, 14.530124308078877429382522211631, 15.12184247945799857599540565439, 16.07580486724887104930693594720, 17.27439423190219804909000545311, 17.98758123615330500366021929914, 18.544381909504664758920616948944, 19.441216125109881320273039696607, 20.67481212684135278315968265443, 20.92489107466448078065620754429, 21.80861106545749948389204728062

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