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Properties

Label 1-283-283.56-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.999 + 0.0359i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (−0.359 − 0.933i)2-s + (−0.441 − 0.897i)3-s + (−0.741 + 0.670i)4-s + (0.380 − 0.924i)5-s + (−0.679 + 0.734i)6-s + (−0.253 + 0.967i)7-s + (0.892 + 0.451i)8-s + (−0.610 + 0.791i)9-s + (−0.999 − 0.0222i)10-s + (0.951 − 0.306i)11-s + (0.929 + 0.369i)12-s + (0.975 + 0.220i)13-s + (0.993 − 0.111i)14-s + (−0.997 + 0.0667i)15-s + (0.100 − 0.994i)16-s + (−0.400 − 0.916i)17-s + ⋯
L(s)  = 1  + (−0.359 − 0.933i)2-s + (−0.441 − 0.897i)3-s + (−0.741 + 0.670i)4-s + (0.380 − 0.924i)5-s + (−0.679 + 0.734i)6-s + (−0.253 + 0.967i)7-s + (0.892 + 0.451i)8-s + (−0.610 + 0.791i)9-s + (−0.999 − 0.0222i)10-s + (0.951 − 0.306i)11-s + (0.929 + 0.369i)12-s + (0.975 + 0.220i)13-s + (0.993 − 0.111i)14-s + (−0.997 + 0.0667i)15-s + (0.100 − 0.994i)16-s + (−0.400 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.999 + 0.0359i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.999 + 0.0359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02074541899 - 1.152198678i\)
\(L(\frac12)\) \(\approx\) \(0.02074541899 - 1.152198678i\)
\(L(1)\) \(\approx\) \(0.5427499620 - 0.6009462440i\)
\(L(1)\) \(\approx\) \(0.5427499620 - 0.6009462440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.359 - 0.933i)T \)
3 \( 1 + (-0.441 - 0.897i)T \)
5 \( 1 + (0.380 - 0.924i)T \)
7 \( 1 + (-0.253 + 0.967i)T \)
11 \( 1 + (0.951 - 0.306i)T \)
13 \( 1 + (0.975 + 0.220i)T \)
17 \( 1 + (-0.400 - 0.916i)T \)
19 \( 1 + (0.296 + 0.955i)T \)
23 \( 1 + (-0.122 - 0.992i)T \)
29 \( 1 + (0.231 - 0.972i)T \)
31 \( 1 + (-0.984 - 0.177i)T \)
37 \( 1 + (0.970 - 0.242i)T \)
41 \( 1 + (0.836 + 0.547i)T \)
43 \( 1 + (0.0334 - 0.999i)T \)
47 \( 1 + (0.679 + 0.734i)T \)
53 \( 1 + (-0.100 - 0.994i)T \)
59 \( 1 + (0.0111 + 0.999i)T \)
61 \( 1 + (0.480 - 0.876i)T \)
67 \( 1 + (-0.784 - 0.619i)T \)
71 \( 1 + (-0.741 - 0.670i)T \)
73 \( 1 + (0.984 - 0.177i)T \)
79 \( 1 + (0.0334 + 0.999i)T \)
83 \( 1 + (0.317 + 0.948i)T \)
89 \( 1 + (-0.848 - 0.528i)T \)
97 \( 1 + (-0.798 + 0.602i)T \)
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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

−25.97496819707556961332460749317, −25.28814429737023185874590367822, −23.71106885917779764998885199928, −23.23996459050022999823158131935, −22.244548255177172932886122229839, −21.75679114470068828033919394189, −20.20108932537041430479337652821, −19.43459202334412846793635230187, −17.981301960889829850070096853352, −17.53504768642819443801470197122, −16.64550424760351603883174370147, −15.7332128864289095195215922173, −14.87310484923792037045892642214, −14.111656583450024747708726870071, −13.131815068947279641758867534526, −11.20683033472161519691091755615, −10.626952390567614533152318315913, −9.685045628550829194481515599158, −8.88939748230765984136850152032, −7.336830527720572347170699006571, −6.50421021679092542511586748692, −5.722739953765414332718409640883, −4.29966722532358709048918227131, −3.49799569253180547528965027101, −1.18123250273803828249910023471, 0.505286071781092755577044571171, 1.53787145724711193219782271763, 2.5348986878626763556741356159, 4.12464074279573829872935143240, 5.46817339220333972332203220728, 6.36511855700538439335477451044, 7.98521240600022467367448407594, 8.82763157057503583319353635223, 9.536380694845809972249065318349, 11.06426302883870027562132456169, 11.894235938339574057144738572433, 12.49844833137304419271437187960, 13.38938466600476433813517577952, 14.20558239445495913915464708256, 16.21143376403512319329566927794, 16.749859426628296859178953260923, 17.92405363969353218943541196912, 18.47210905676484188368139470059, 19.345074599263010557216945918054, 20.29042644034140335241324997278, 21.1449658392209849246304552468, 22.21934765286834408676322830387, 22.81075761193503320831559318605, 24.09151433913684563023025936340, 25.04767217409532726416239825564

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