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Properties

Label 1-1148-1148.215-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.948 + 0.318i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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)3-s + (−0.913 − 0.406i)5-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)25-s + 27-s + (−0.809 + 0.587i)29-s + (0.913 − 0.406i)31-s + (0.104 − 0.994i)33-s + (0.913 + 0.406i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.913 − 0.406i)5-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)25-s + 27-s + (−0.809 + 0.587i)29-s + (0.913 − 0.406i)31-s + (0.104 − 0.994i)33-s + (0.913 + 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.948 + 0.318i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.948 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5730284375 + 0.09361779818i\)
\(L(\frac12)\) \(\approx\) \(0.5730284375 + 0.09361779818i\)
\(L(1)\) \(\approx\) \(0.6086684756 + 0.1056944264i\)
\(L(1)\) \(\approx\) \(0.6086684756 + 0.1056944264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−21.405631961233906713533855041072, −20.273374081640769173631971496828, −19.52081382535262971658946297079, −18.85060138816834043452736490543, −18.38254971457413572428828279809, −17.50336255589950512859077272840, −16.578614924121360317650923324933, −15.96101598887953403640337981271, −15.07279215841974016710787803431, −14.135588631252198394563984868264, −13.37663719136087064786173908526, −12.54586064892645309521724277531, −11.82057729880393514310196971684, −11.06767929106775684068409699082, −10.58093490393480900506077273043, −9.18713596168769435304417181460, −8.15967019563588006696729451108, −7.66150151904826652777814278488, −6.66853937582376932385606395948, −6.18015900766563198123101942169, −4.85147556777401422539925865306, −4.18803330181395817329151461547, −2.745929591897257204502057041181, −2.13885332437850062984349371903, −0.54897038475337731560646117063, 0.487695697561938024922745929, 2.21339722345352359548566717420, 3.42167969930515474346113629932, 4.18227811447923396014235850291, 4.96476626632411682354609893893, 5.69189492310797003824375713397, 6.80493074225679106154079542507, 7.906698821644442103307837222090, 8.484819436041596633992464825051, 9.54155602970962887073711483087, 10.384205710338812305042084657606, 10.9923775306012152890713773894, 11.84428993204472884331998879298, 12.63210839985494332168638319225, 13.30332744005404601685430236997, 14.763342645355673844327691962126, 15.34660875311202118466100362661, 15.71380505388764733574800461500, 16.675553164389874653691616805454, 17.34813136753434089711807868360, 18.106070207147594607355493896170, 19.10998016709818047217593908342, 20.14371127169176244332820231315, 20.38801328191082236943562978159, 21.36640215696345877498631426216

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