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Properties

Label 1-6145-6145.1058-r0-0-0
Degree $1$
Conductor $6145$
Sign $0.472 - 0.881i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
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  + (−0.580 − 0.814i)2-s + (0.502 − 0.864i)3-s + (−0.326 + 0.945i)4-s + (−0.995 + 0.0919i)6-s + (−0.507 + 0.861i)7-s + (0.959 − 0.282i)8-s + (−0.494 − 0.869i)9-s + (−0.783 + 0.621i)11-s + (0.652 + 0.757i)12-s + (−0.856 + 0.516i)13-s + (0.996 − 0.0868i)14-s + (−0.786 − 0.617i)16-s + (−0.940 + 0.340i)17-s + (−0.421 + 0.906i)18-s + (0.0306 − 0.999i)19-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (0.502 − 0.864i)3-s + (−0.326 + 0.945i)4-s + (−0.995 + 0.0919i)6-s + (−0.507 + 0.861i)7-s + (0.959 − 0.282i)8-s + (−0.494 − 0.869i)9-s + (−0.783 + 0.621i)11-s + (0.652 + 0.757i)12-s + (−0.856 + 0.516i)13-s + (0.996 − 0.0868i)14-s + (−0.786 − 0.617i)16-s + (−0.940 + 0.340i)17-s + (−0.421 + 0.906i)18-s + (0.0306 − 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5319752965 - 0.3184594073i\)
\(L(\frac12)\) \(\approx\) \(0.5319752965 - 0.3184594073i\)
\(L(1)\) \(\approx\) \(0.5685109529 - 0.2999685124i\)
\(L(1)\) \(\approx\) \(0.5685109529 - 0.2999685124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.580 - 0.814i)T \)
3 \( 1 + (0.502 - 0.864i)T \)
7 \( 1 + (-0.507 + 0.861i)T \)
11 \( 1 + (-0.783 + 0.621i)T \)
13 \( 1 + (-0.856 + 0.516i)T \)
17 \( 1 + (-0.940 + 0.340i)T \)
19 \( 1 + (0.0306 - 0.999i)T \)
23 \( 1 + (0.0664 - 0.997i)T \)
29 \( 1 + (-0.444 + 0.895i)T \)
31 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (0.989 - 0.142i)T \)
41 \( 1 + (-0.946 + 0.321i)T \)
43 \( 1 + (-0.999 + 0.0255i)T \)
47 \( 1 + (0.122 - 0.992i)T \)
53 \( 1 + (0.262 + 0.964i)T \)
59 \( 1 + (-0.307 + 0.951i)T \)
61 \( 1 + (0.297 - 0.954i)T \)
67 \( 1 + (-0.360 - 0.932i)T \)
71 \( 1 + (0.808 + 0.588i)T \)
73 \( 1 + (0.934 - 0.355i)T \)
79 \( 1 + (-0.959 - 0.282i)T \)
83 \( 1 + (0.277 + 0.960i)T \)
89 \( 1 + (-0.257 - 0.966i)T \)
97 \( 1 + (0.831 + 0.555i)T \)
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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

−17.55041401314568534994580488639, −16.99600521110305104013773356768, −16.38669758763412374753001074670, −15.91190823690152927718699264988, −15.30909586428321947040805063284, −14.69914244657928110179524018923, −13.99961508543527597588673673230, −13.42087969000973751459253158234, −12.93306119906143305790631130714, −11.52570619658309596547578573694, −10.88742859179347779527676536171, −10.21760791928908637583349431115, −9.82458876334856559616086329587, −9.2187856905840488391308796009, −8.350596716429191713951060634536, −7.8162067990466725706934399607, −7.27933778521859868513079027860, −6.386244872067486463670417430537, −5.51025372122959945917842107567, −5.047736275729757027832833593115, −4.18180742081836033299138908009, −3.47823476236066734458302064481, −2.62811171900313289700656673900, −1.67025427861072879306518076088, −0.368045378690619827414812590081, 0.421027170833228002698477649000, 1.7458339174978584147630929789, 2.30199848491676097836843164534, 2.66398275502964029221177713854, 3.516102484350772875825796272331, 4.5163441991625161789428567559, 5.217021458954842147290851689876, 6.4122022403937383941477381166, 6.97162310241425966102947095832, 7.585087484388624218922176611382, 8.409763117773822325629833038821, 8.96742561384977022942682541553, 9.44193727508466683434075425542, 10.17496096770014878417455235267, 11.0723162075178658303584474623, 11.7164423408376637194765755703, 12.37243787411230571091325076663, 12.93805893895929074734132829988, 13.20411681259893031740736725535, 14.13369984019420661425548374699, 15.09149535069831728417780685968, 15.382160049173301214974389731432, 16.57845183643142231077052229922, 17.000334394405997554839833815870, 18.00143214487960113419836568395

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