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Properties

Label 1-283-283.168-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.0851 + 0.996i$
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  + (−0.0334 + 0.999i)2-s + (−0.420 − 0.907i)3-s + (−0.997 − 0.0667i)4-s + (−0.645 − 0.763i)5-s + (0.920 − 0.390i)6-s + (−0.166 + 0.986i)7-s + (0.100 − 0.994i)8-s + (−0.645 + 0.763i)9-s + (0.784 − 0.619i)10-s + (−0.997 + 0.0667i)11-s + (0.359 + 0.933i)12-s + (0.920 − 0.390i)13-s + (−0.979 − 0.199i)14-s + (−0.420 + 0.907i)15-s + (0.991 + 0.133i)16-s + (−0.979 + 0.199i)17-s + ⋯
L(s)  = 1  + (−0.0334 + 0.999i)2-s + (−0.420 − 0.907i)3-s + (−0.997 − 0.0667i)4-s + (−0.645 − 0.763i)5-s + (0.920 − 0.390i)6-s + (−0.166 + 0.986i)7-s + (0.100 − 0.994i)8-s + (−0.645 + 0.763i)9-s + (0.784 − 0.619i)10-s + (−0.997 + 0.0667i)11-s + (0.359 + 0.933i)12-s + (0.920 − 0.390i)13-s + (−0.979 − 0.199i)14-s + (−0.420 + 0.907i)15-s + (0.991 + 0.133i)16-s + (−0.979 + 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0851 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0851 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0851 + 0.996i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.0851 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3918160830 + 0.4267369176i\)
\(L(\frac12)\) \(\approx\) \(0.3918160830 + 0.4267369176i\)
\(L(1)\) \(\approx\) \(0.6245829025 + 0.2043242128i\)
\(L(1)\) \(\approx\) \(0.6245829025 + 0.2043242128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.0334 + 0.999i)T \)
3 \( 1 + (-0.420 - 0.907i)T \)
5 \( 1 + (-0.645 - 0.763i)T \)
7 \( 1 + (-0.166 + 0.986i)T \)
11 \( 1 + (-0.997 + 0.0667i)T \)
13 \( 1 + (0.920 - 0.390i)T \)
17 \( 1 + (-0.979 + 0.199i)T \)
19 \( 1 + (0.920 - 0.390i)T \)
23 \( 1 + (0.860 + 0.509i)T \)
29 \( 1 + (-0.741 + 0.670i)T \)
31 \( 1 + (0.593 + 0.805i)T \)
37 \( 1 + (0.480 + 0.876i)T \)
41 \( 1 + (0.100 + 0.994i)T \)
43 \( 1 + (-0.538 + 0.842i)T \)
47 \( 1 + (0.920 + 0.390i)T \)
53 \( 1 + (0.991 - 0.133i)T \)
59 \( 1 + (-0.944 - 0.328i)T \)
61 \( 1 + (0.784 + 0.619i)T \)
67 \( 1 + (0.359 - 0.933i)T \)
71 \( 1 + (-0.997 + 0.0667i)T \)
73 \( 1 + (0.593 - 0.805i)T \)
79 \( 1 + (-0.538 - 0.842i)T \)
83 \( 1 + (0.964 + 0.264i)T \)
89 \( 1 + (-0.538 + 0.842i)T \)
97 \( 1 + (0.860 + 0.509i)T \)
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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

−26.04045667605388143338282091461, −23.9356017699675513081497726275, −23.0039897026746126515011119333, −22.7563332910542442976984170175, −21.68591599491143147075909354499, −20.63367643866790893318545683079, −20.26196590746526742824331907131, −18.96360345730344985736638309, −18.23018250764513285495868233947, −17.187634097740617257674757448216, −16.12116869089656125578156848591, −15.2345446017964373745175823558, −14.04933524709022867291959253857, −13.2333335254100384834017662923, −11.82248977857135336987194211151, −10.98262193102406689004923831573, −10.5943850071876207783077006480, −9.58989331591709245712003565517, −8.39844514338720517814714437714, −7.13233924240500267468911950860, −5.64882010785296441311960767599, −4.30792940725975805507470776818, −3.73554242551967040489527038764, −2.63031299356510902117182634017, −0.49638736649750058166976080289, 1.17192134590011584898581382132, 3.0750263920642578628481818283, 4.86047855865837125484570454148, 5.49109769864541336043316034397, 6.550590575713222622949841193202, 7.67871873998091407112803930192, 8.403720073771071096721682887329, 9.21755060713802559091732605648, 10.95059185426465619617377499698, 12.04057012970033180721566906506, 13.112148566385225122202562917705, 13.34985390515065288015396370085, 15.068835633930143845625273879282, 15.77809373946632070895016255973, 16.48165852949540598316163651668, 17.71349815722120604121370708727, 18.28382325874228628597878788511, 19.11180209194936867624861353392, 20.16482645347450370556270819077, 21.56091487055149958771985128275, 22.63421393203547275300819289777, 23.37659511458179324005942260058, 24.071021378058070367901260695705, 24.8121594430388594734308131062, 25.47829322109225352926825488164

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