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Properties

Label 1-283-283.222-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.543 - 0.839i$
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  + (−0.991 + 0.133i)2-s + (0.166 − 0.986i)3-s + (0.964 − 0.264i)4-s + (0.944 − 0.328i)5-s + (−0.0334 + 0.999i)6-s + (0.784 − 0.619i)7-s + (−0.920 + 0.390i)8-s + (−0.944 − 0.328i)9-s + (−0.892 + 0.451i)10-s + (0.964 + 0.264i)11-s + (−0.100 − 0.994i)12-s + (−0.0334 + 0.999i)13-s + (−0.695 + 0.718i)14-s + (−0.166 − 0.986i)15-s + (0.860 − 0.509i)16-s + (−0.695 − 0.718i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.133i)2-s + (0.166 − 0.986i)3-s + (0.964 − 0.264i)4-s + (0.944 − 0.328i)5-s + (−0.0334 + 0.999i)6-s + (0.784 − 0.619i)7-s + (−0.920 + 0.390i)8-s + (−0.944 − 0.328i)9-s + (−0.892 + 0.451i)10-s + (0.964 + 0.264i)11-s + (−0.100 − 0.994i)12-s + (−0.0334 + 0.999i)13-s + (−0.695 + 0.718i)14-s + (−0.166 − 0.986i)15-s + (0.860 − 0.509i)16-s + (−0.695 − 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.543 - 0.839i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7625206310 - 1.401699473i\)
\(L(\frac12)\) \(\approx\) \(0.7625206310 - 1.401699473i\)
\(L(1)\) \(\approx\) \(0.8506357002 - 0.4698445584i\)
\(L(1)\) \(\approx\) \(0.8506357002 - 0.4698445584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.991 + 0.133i)T \)
3 \( 1 + (0.166 - 0.986i)T \)
5 \( 1 + (0.944 - 0.328i)T \)
7 \( 1 + (0.784 - 0.619i)T \)
11 \( 1 + (0.964 + 0.264i)T \)
13 \( 1 + (-0.0334 + 0.999i)T \)
17 \( 1 + (-0.695 - 0.718i)T \)
19 \( 1 + (0.0334 - 0.999i)T \)
23 \( 1 + (-0.538 - 0.842i)T \)
29 \( 1 + (-0.979 + 0.199i)T \)
31 \( 1 + (0.824 - 0.565i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
41 \( 1 + (0.920 + 0.390i)T \)
43 \( 1 + (0.645 + 0.763i)T \)
47 \( 1 + (0.0334 + 0.999i)T \)
53 \( 1 + (-0.860 - 0.509i)T \)
59 \( 1 + (0.231 - 0.972i)T \)
61 \( 1 + (-0.892 - 0.451i)T \)
67 \( 1 + (-0.100 + 0.994i)T \)
71 \( 1 + (0.964 + 0.264i)T \)
73 \( 1 + (-0.824 - 0.565i)T \)
79 \( 1 + (0.645 - 0.763i)T \)
83 \( 1 + (0.480 - 0.876i)T \)
89 \( 1 + (-0.645 - 0.763i)T \)
97 \( 1 + (-0.538 - 0.842i)T \)
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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

−25.70420922709414374700084778699, −25.10653980510247496207202669309, −24.341542494268587660389224236790, −22.507686477739561319387290091627, −21.7931527987405854439178175604, −21.08892735838267094058923378891, −20.29389350422874973953550083409, −19.3178219652472872249278251785, −18.157484038837135330147341024369, −17.40243304664861431760590571705, −16.78583046182760842022196519880, −15.45545756256079268580145995952, −14.89486153949247384001252452762, −13.85819786084490418159025979538, −12.23997010494466585415730798463, −11.17998884298212549291952536022, −10.44573983589408953766569518180, −9.55048189090432263884443808669, −8.76318561982018438799381337119, −7.87101264286503992908679567595, −6.20320078702582036658581606171, −5.50968172655711549670660800257, −3.79747394813905279258866193726, −2.57576963943311204636830842952, −1.49455012657445917391134366497, 0.66255143589311791659923169174, 1.66782061704955853349645853897, 2.4514817922058435730972044882, 4.57747143684091462970477011212, 6.145991180040111008560173937242, 6.830087806114661396906846371261, 7.77383447934355283813056007200, 8.97969422773560398698234411679, 9.45708555021183904284465515780, 11.01741950439607689700201778912, 11.669819888986883214188660418545, 12.91239646220708975912117374055, 14.09723473033015119317833581628, 14.55484582651678284422273750696, 16.24772148802702232601012102243, 17.2193678499019753850116719027, 17.63504839241679812952450793194, 18.450776654728218785107304479100, 19.539897819315811091317811266009, 20.2879791001323191018355236332, 21.02000975080286333032340126129, 22.342281026976421493337521770436, 23.77846977167376009052653338760, 24.4946376826108802953918042954, 24.84849460714694950961343715514

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