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Properties

Label 1-283-283.31-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.399 - 0.916i$
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  + (−0.480 − 0.876i)2-s + (−0.726 − 0.687i)3-s + (−0.538 + 0.842i)4-s + (−0.836 − 0.547i)5-s + (−0.253 + 0.967i)6-s + (−0.993 − 0.111i)7-s + (0.997 + 0.0667i)8-s + (0.0556 + 0.998i)9-s + (−0.0779 + 0.996i)10-s + (−0.460 + 0.887i)11-s + (0.970 − 0.242i)12-s + (−0.711 − 0.703i)13-s + (0.380 + 0.924i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.610 + 0.791i)17-s + ⋯
L(s)  = 1  + (−0.480 − 0.876i)2-s + (−0.726 − 0.687i)3-s + (−0.538 + 0.842i)4-s + (−0.836 − 0.547i)5-s + (−0.253 + 0.967i)6-s + (−0.993 − 0.111i)7-s + (0.997 + 0.0667i)8-s + (0.0556 + 0.998i)9-s + (−0.0779 + 0.996i)10-s + (−0.460 + 0.887i)11-s + (0.970 − 0.242i)12-s + (−0.711 − 0.703i)13-s + (0.380 + 0.924i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.610 + 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.399 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1659241165 - 0.2531757736i\)
\(L(\frac12)\) \(\approx\) \(0.1659241165 - 0.2531757736i\)
\(L(1)\) \(\approx\) \(0.3403605318 - 0.2107202486i\)
\(L(1)\) \(\approx\) \(0.3403605318 - 0.2107202486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.480 - 0.876i)T \)
3 \( 1 + (-0.726 - 0.687i)T \)
5 \( 1 + (-0.836 - 0.547i)T \)
7 \( 1 + (-0.993 - 0.111i)T \)
11 \( 1 + (-0.460 + 0.887i)T \)
13 \( 1 + (-0.711 - 0.703i)T \)
17 \( 1 + (0.610 + 0.791i)T \)
19 \( 1 + (-0.964 + 0.264i)T \)
23 \( 1 + (0.937 + 0.348i)T \)
29 \( 1 + (-0.0334 + 0.999i)T \)
31 \( 1 + (-0.811 - 0.584i)T \)
37 \( 1 + (-0.756 - 0.654i)T \)
41 \( 1 + (0.441 - 0.897i)T \)
43 \( 1 + (-0.784 + 0.619i)T \)
47 \( 1 + (0.253 + 0.967i)T \)
53 \( 1 + (0.420 - 0.907i)T \)
59 \( 1 + (-0.679 + 0.734i)T \)
61 \( 1 + (-0.824 + 0.565i)T \)
67 \( 1 + (-0.695 + 0.718i)T \)
71 \( 1 + (-0.538 - 0.842i)T \)
73 \( 1 + (0.811 - 0.584i)T \)
79 \( 1 + (-0.784 - 0.619i)T \)
83 \( 1 + (-0.338 - 0.940i)T \)
89 \( 1 + (-0.929 - 0.369i)T \)
97 \( 1 + (-0.770 + 0.637i)T \)
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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

−25.958970315734241114542955062770, −24.73755962635694919792126886152, −23.561507103999462800853700594831, −23.171083924175020952261904136881, −22.28718274779215021652772495970, −21.43100988800113138086660549995, −19.88273736557968764626542361922, −18.91665788155536147044118581912, −18.470562823923364182479217112490, −16.92839607882115969017595792794, −16.53311451883568172368563629853, −15.60656909732050956836572025492, −15.01990034582020172449520563016, −13.86232059596213925191848192985, −12.458300645792058429055734343908, −11.29338033692343785388001338873, −10.43850537321865767753742011542, −9.53310981340341360440516146688, −8.53692186107641984382643888650, −7.137552857341078015275457083570, −6.49883855076608575344033880776, −5.370992124850898851398416165463, −4.28304112316022172028441972975, −3.04333320907211513184314991535, −0.444567786634776762952575460791, 0.30323870412106616217580890721, 1.64756308421329198663682425578, 3.062590269163289076188514901742, 4.33260525062601906551793865975, 5.47017455054071240047330763848, 7.14075340666073082197030292833, 7.73340238974451676907334931138, 8.9426005051155038932761711512, 10.2004048358582270968579274152, 10.89530142101356952966053664754, 12.24965073394262167893832091739, 12.59345961462526499874188618935, 13.17056204064260692548446054843, 14.9942105019060723041779037938, 16.29529813310696141932738024286, 16.960761520583159359771275036524, 17.7957716270811954966459335751, 18.97687610804441858700672793584, 19.4351578318834598201359967306, 20.23437987204842367509102288613, 21.38373522758978008815027803976, 22.59810765919192117051053723275, 23.02414528820674415641563840678, 23.921948479292041324773611535237, 25.26324670502435515387342506318

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