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Properties

Label 1-283-283.4-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.888 - 0.459i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (−0.296 + 0.955i)2-s + (0.695 + 0.718i)3-s + (−0.824 − 0.565i)4-s + (−0.0334 − 0.999i)5-s + (−0.892 + 0.451i)6-s + (−0.997 + 0.0667i)7-s + (0.784 − 0.619i)8-s + (−0.0334 + 0.999i)9-s + (0.964 + 0.264i)10-s + (−0.824 + 0.565i)11-s + (−0.166 − 0.986i)12-s + (−0.892 + 0.451i)13-s + (0.231 − 0.972i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.231 + 0.972i)17-s + ⋯
L(s)  = 1  + (−0.296 + 0.955i)2-s + (0.695 + 0.718i)3-s + (−0.824 − 0.565i)4-s + (−0.0334 − 0.999i)5-s + (−0.892 + 0.451i)6-s + (−0.997 + 0.0667i)7-s + (0.784 − 0.619i)8-s + (−0.0334 + 0.999i)9-s + (0.964 + 0.264i)10-s + (−0.824 + 0.565i)11-s + (−0.166 − 0.986i)12-s + (−0.892 + 0.451i)13-s + (0.231 − 0.972i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.231 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.888 - 0.459i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.888 - 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1067632024 + 0.4389113692i\)
\(L(\frac12)\) \(\approx\) \(-0.1067632024 + 0.4389113692i\)
\(L(1)\) \(\approx\) \(0.5313082209 + 0.4514341297i\)
\(L(1)\) \(\approx\) \(0.5313082209 + 0.4514341297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.296 + 0.955i)T \)
3 \( 1 + (0.695 + 0.718i)T \)
5 \( 1 + (-0.0334 - 0.999i)T \)
7 \( 1 + (-0.997 + 0.0667i)T \)
11 \( 1 + (-0.824 + 0.565i)T \)
13 \( 1 + (-0.892 + 0.451i)T \)
17 \( 1 + (0.231 + 0.972i)T \)
19 \( 1 + (-0.892 + 0.451i)T \)
23 \( 1 + (0.100 - 0.994i)T \)
29 \( 1 + (-0.944 + 0.328i)T \)
31 \( 1 + (-0.538 + 0.842i)T \)
37 \( 1 + (-0.979 - 0.199i)T \)
41 \( 1 + (0.784 + 0.619i)T \)
43 \( 1 + (0.920 + 0.390i)T \)
47 \( 1 + (-0.892 - 0.451i)T \)
53 \( 1 + (0.359 - 0.933i)T \)
59 \( 1 + (0.991 - 0.133i)T \)
61 \( 1 + (0.964 - 0.264i)T \)
67 \( 1 + (-0.166 + 0.986i)T \)
71 \( 1 + (-0.824 + 0.565i)T \)
73 \( 1 + (-0.538 - 0.842i)T \)
79 \( 1 + (0.920 - 0.390i)T \)
83 \( 1 + (-0.741 + 0.670i)T \)
89 \( 1 + (0.920 + 0.390i)T \)
97 \( 1 + (0.100 - 0.994i)T \)
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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

−25.52101430415051406976043188024, −24.076044472472909953615651918159, −23.00713588241231203976377539148, −22.341071626780674374319458810201, −21.30207464357412316292780605527, −20.335912097225727534610007427045, −19.2092985042755527704325824680, −19.10678378623048521347234371870, −18.11360742697427633199790726553, −17.2166493312088630091208576223, −15.7207292887713308361361346999, −14.61286814154765168020511471330, −13.558969185101568691416177102393, −13.0220917596930191901184254817, −11.96846912678214671423008508054, −10.88916510601430228198311095795, −9.88122282694290023521441651068, −9.11147451489013225900007745058, −7.73362276423295347291568778843, −7.15411314121487051097801113675, −5.67251992003287807557228967816, −3.76917063646135976505860189482, −2.89090257681656494394843551205, −2.28995461543096473330613256197, −0.28437608045897218871164099874, 2.05418725897829134178458605987, 3.80630786518058094688161870269, 4.7239829527381604619092600492, 5.66207016305856774380808305417, 7.061853233839785765394037953201, 8.168689579171604598054127052587, 8.9188077095362055188418156014, 9.79693792794093447969409970034, 10.46119085823778586632011819633, 12.66837995474138443321971330766, 13.036289766942549724954767811681, 14.448519615033184436043469685815, 15.09212230963183236661730228713, 16.24170018016362205426220651962, 16.47122148871053847287721410979, 17.56252388724804512994477001179, 19.07083592856328852784890019578, 19.52157666846902012948920794217, 20.626330531345759704181922375527, 21.597731329228545324381558870144, 22.59392119542603837858739473453, 23.578436603817522085840356047677, 24.50662107255188594856942493554, 25.35866776522834220441011286598, 26.041089012569910089920182703393

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