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Properties

Label 1-1045-1045.677-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.842 + 0.539i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.994 − 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s − i·12-s + (0.406 − 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.951 − 0.309i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.104 − 0.994i)24-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s − i·12-s + (0.406 − 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.951 − 0.309i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.104 − 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.842 + 0.539i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.842 + 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6040111692 - 2.063441041i\)
\(L(\frac12)\) \(\approx\) \(-0.6040111692 - 2.063441041i\)
\(L(1)\) \(\approx\) \(1.335634874 - 0.9745550074i\)
\(L(1)\) \(\approx\) \(1.335634874 - 0.9745550074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.743 + 0.669i)T \)
53 \( 1 + (-0.406 + 0.913i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−21.942928841779859993851778324172, −21.21750070895366406222028499264, −20.48700418703519918897215097547, −19.61133957203847053186985706517, −19.13506840459056101106215548859, −17.69092308376761604705481495156, −16.60540071432115922955373976778, −16.21178509472907611193592784904, −15.47692723215060391645226342667, −14.82803432114735659519545842068, −13.90035333689991261224023645359, −13.3626974143261472101923288743, −12.22727219322363658674944293554, −11.67523104610256993893614358668, −10.52136319008996176115427637819, −10.08037511207578687153171182202, −8.858973311924496400540343094665, −8.21688441957478567465373641885, −6.69225213427933600535535681997, −6.27347599264600826945442864916, −5.21609469709922140574750010101, −4.33938203683398343910376522629, −3.62313093341697059178975214478, −2.83808252081695072511843221294, −1.79923371275961534349043981655, 0.26839689313806019536476195781, 1.32537721945355824886060104388, 2.56375806121206448759440967011, 3.14296767776878092718693813801, 4.11207673406388381610966142256, 5.39930472971950826960748060836, 6.141254504642534763820121558744, 6.871746574587456142670240694631, 7.58568298879312071755156375238, 8.56535460258118635180170614870, 9.816987258728948318149464995801, 10.62460984591134097753295490860, 11.738767346003522840328978051595, 12.23728404491050141263866443751, 13.17962785871789945227943414072, 13.58593847792027543201486747790, 14.22570226933635573683864190979, 15.45163198929947931601252576674, 15.85741046266179116930769036512, 16.96706161175677778275064901273, 17.78992911805890496322821694949, 18.807136954296237186748852863404, 19.46354449070765578727119371115, 20.2521427995404849949936671078, 20.61142818167350215803718054005

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