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Properties

Label 1-283-283.151-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.287 + 0.957i$
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  + (0.920 + 0.390i)2-s + (0.480 + 0.876i)3-s + (0.695 + 0.718i)4-s + (−0.538 − 0.842i)5-s + (0.100 + 0.994i)6-s + (−0.420 + 0.907i)7-s + (0.359 + 0.933i)8-s + (−0.538 + 0.842i)9-s + (−0.166 − 0.986i)10-s + (0.695 − 0.718i)11-s + (−0.296 + 0.955i)12-s + (0.100 + 0.994i)13-s + (−0.741 + 0.670i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (−0.741 − 0.670i)17-s + ⋯
L(s)  = 1  + (0.920 + 0.390i)2-s + (0.480 + 0.876i)3-s + (0.695 + 0.718i)4-s + (−0.538 − 0.842i)5-s + (0.100 + 0.994i)6-s + (−0.420 + 0.907i)7-s + (0.359 + 0.933i)8-s + (−0.538 + 0.842i)9-s + (−0.166 − 0.986i)10-s + (0.695 − 0.718i)11-s + (−0.296 + 0.955i)12-s + (0.100 + 0.994i)13-s + (−0.741 + 0.670i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (−0.741 − 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.294360125 + 1.740312954i\)
\(L(\frac12)\) \(\approx\) \(1.294360125 + 1.740312954i\)
\(L(1)\) \(\approx\) \(1.495039845 + 0.9964504378i\)
\(L(1)\) \(\approx\) \(1.495039845 + 0.9964504378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.920 + 0.390i)T \)
3 \( 1 + (0.480 + 0.876i)T \)
5 \( 1 + (-0.538 - 0.842i)T \)
7 \( 1 + (-0.420 + 0.907i)T \)
11 \( 1 + (0.695 - 0.718i)T \)
13 \( 1 + (0.100 + 0.994i)T \)
17 \( 1 + (-0.741 - 0.670i)T \)
19 \( 1 + (0.100 + 0.994i)T \)
23 \( 1 + (0.991 + 0.133i)T \)
29 \( 1 + (-0.824 - 0.565i)T \)
31 \( 1 + (0.231 - 0.972i)T \)
37 \( 1 + (0.964 + 0.264i)T \)
41 \( 1 + (0.359 - 0.933i)T \)
43 \( 1 + (0.860 + 0.509i)T \)
47 \( 1 + (0.100 - 0.994i)T \)
53 \( 1 + (-0.0334 - 0.999i)T \)
59 \( 1 + (-0.645 - 0.763i)T \)
61 \( 1 + (-0.166 + 0.986i)T \)
67 \( 1 + (-0.296 - 0.955i)T \)
71 \( 1 + (0.695 - 0.718i)T \)
73 \( 1 + (0.231 + 0.972i)T \)
79 \( 1 + (0.860 - 0.509i)T \)
83 \( 1 + (-0.997 - 0.0667i)T \)
89 \( 1 + (0.860 + 0.509i)T \)
97 \( 1 + (0.991 + 0.133i)T \)
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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

−25.20932634449271050295537884155, −24.19615453433133580086916701979, −23.315661867436556558210743952121, −22.813878571663100513279173782691, −21.958827853864676472954260871804, −20.473109723240664138244387513518, −19.78947180814927661430089483315, −19.428762527242289799396858586742, −18.20269962120822617033562896073, −17.203647913634844933520844319596, −15.60769942594744037698886566133, −14.88628743286576463794512736634, −14.12253001334998730631565172931, −13.09120435939012783789643080870, −12.54790763703658043986284420239, −11.27321794575769918574095602968, −10.60275318807962924520121287413, −9.2640291977026398530477124556, −7.59496749361918701415987664676, −6.95738122890196088078484319769, −6.195931692507276875098067136132, −4.432793271203412183889106456578, −3.41531540505454408162872363023, −2.60131330199884489012204873382, −1.1019014304225845250392959380, 2.19741624683429949256053869054, 3.493837857282057261728201982, 4.250511885145134233667330386457, 5.26873326826454163513775542434, 6.25781069081617018557155485608, 7.74981124484922508070213085337, 8.81833392290336697852418169859, 9.36348497679880333673202007685, 11.271732578636295438765642246396, 11.7626115450022739341775122279, 12.98115429985028796188445671221, 13.877845383864817514020931118656, 14.89699484600887502742944230331, 15.64271715052123430906420851340, 16.43931358813341052112331349051, 16.90691802690938589193127670198, 18.888609551032948796737059349836, 19.71717964968189743817542736499, 20.75949791259073929416869289761, 21.30605007499128451200882631568, 22.27233770368804430142162496339, 22.9144443811962018385923139170, 24.30714777339402328420770315362, 24.75406357577182770500127819312, 25.64635081911710330011887746051

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