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Properties

Label 1-283-283.160-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.931 - 0.363i$
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  + (−0.741 − 0.670i)2-s + (−0.610 − 0.791i)3-s + (0.100 + 0.994i)4-s + (−0.711 + 0.703i)5-s + (−0.0779 + 0.996i)6-s + (−0.871 + 0.490i)7-s + (0.593 − 0.805i)8-s + (−0.253 + 0.967i)9-s + (0.999 − 0.0445i)10-s + (0.811 + 0.584i)11-s + (0.726 − 0.687i)12-s + (0.902 − 0.431i)13-s + (0.975 + 0.220i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.679 − 0.734i)17-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)2-s + (−0.610 − 0.791i)3-s + (0.100 + 0.994i)4-s + (−0.711 + 0.703i)5-s + (−0.0779 + 0.996i)6-s + (−0.871 + 0.490i)7-s + (0.593 − 0.805i)8-s + (−0.253 + 0.967i)9-s + (0.999 − 0.0445i)10-s + (0.811 + 0.584i)11-s + (0.726 − 0.687i)12-s + (0.902 − 0.431i)13-s + (0.975 + 0.220i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.679 − 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04599804222 - 0.2445598346i\)
\(L(\frac12)\) \(\approx\) \(0.04599804222 - 0.2445598346i\)
\(L(1)\) \(\approx\) \(0.3972071761 - 0.1770465550i\)
\(L(1)\) \(\approx\) \(0.3972071761 - 0.1770465550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.741 - 0.670i)T \)
3 \( 1 + (-0.610 - 0.791i)T \)
5 \( 1 + (-0.711 + 0.703i)T \)
7 \( 1 + (-0.871 + 0.490i)T \)
11 \( 1 + (0.811 + 0.584i)T \)
13 \( 1 + (0.902 - 0.431i)T \)
17 \( 1 + (-0.679 - 0.734i)T \)
19 \( 1 + (-0.824 - 0.565i)T \)
23 \( 1 + (-0.970 - 0.242i)T \)
29 \( 1 + (-0.892 + 0.451i)T \)
31 \( 1 + (0.937 - 0.348i)T \)
37 \( 1 + (0.882 + 0.470i)T \)
41 \( 1 + (0.400 - 0.916i)T \)
43 \( 1 + (-0.997 + 0.0667i)T \)
47 \( 1 + (-0.0779 - 0.996i)T \)
53 \( 1 + (-0.979 - 0.199i)T \)
59 \( 1 + (-0.999 - 0.0222i)T \)
61 \( 1 + (-0.538 + 0.842i)T \)
67 \( 1 + (0.231 - 0.972i)T \)
71 \( 1 + (0.100 - 0.994i)T \)
73 \( 1 + (0.937 + 0.348i)T \)
79 \( 1 + (-0.997 - 0.0667i)T \)
83 \( 1 + (-0.798 - 0.602i)T \)
89 \( 1 + (0.441 - 0.897i)T \)
97 \( 1 + (0.274 + 0.961i)T \)
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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

−26.30418572963478047769752278569, −25.18174070672449333676165223418, −24.04731790315248678493327460049, −23.39319795816229323512474626259, −22.68427868545851896767352263719, −21.47125703575711600726553413745, −20.266895604560355008127125230445, −19.6049461354082077933702402809, −18.678950162844275538987782988289, −17.28405431266778750148898641675, −16.7255451036396297450283517678, −16.053361065084960006180166410193, −15.408784963632255606122579607643, −14.222108794424655386832411539093, −12.921269090070708032895280003183, −11.57119141679341884883620744127, −10.852879946770838067307998347727, −9.72580210005742578019831017006, −8.93985489353627402270430809866, −8.03259525061010223809167616515, −6.436727532453937863082292190042, −6.041355629409977210990274475556, −4.42368534616268337788893655438, −3.75894557208129447136651037025, −1.21373730073310020412161629087, 0.259706647969032478222534837443, 1.97766942556895324602225117776, 3.05587023181362357487333223970, 4.30432197011463849264030022415, 6.29540178784626424180842022394, 6.87443009201602314786910983470, 7.96370854875666597642003371843, 9.01024991446765696958156858577, 10.24887398997696981658460266718, 11.21362924062503152313701264813, 11.881251375939208086405314852254, 12.7190737343608629992645428325, 13.64066336708555244635955989037, 15.30035838474230983767006186680, 16.16738195826980325115510038351, 17.181602651325341857499590016301, 18.24728461484543720836724936605, 18.62008544892034127208639758946, 19.64793551269343177927985744240, 20.12854323148123482558520689108, 21.80424022145773018666086157329, 22.526293653051245842521018139070, 23.013488072412059175611868617575, 24.38197282999694269275190098395, 25.47419166497588084221998103032

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