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Properties

Label 1-47-47.24-r0-0-0
Degree $1$
Conductor $47$
Sign $0.998 + 0.0450i$
Analytic cond. $0.218267$
Root an. cond. $0.218267$
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  + (0.962 − 0.269i)2-s + (0.460 + 0.887i)3-s + (0.854 − 0.519i)4-s + (−0.775 − 0.631i)5-s + (0.682 + 0.730i)6-s + (−0.990 + 0.136i)7-s + (0.682 − 0.730i)8-s + (−0.576 + 0.816i)9-s + (−0.917 − 0.398i)10-s + (−0.0682 + 0.997i)11-s + (0.854 + 0.519i)12-s + (−0.334 − 0.942i)13-s + (−0.917 + 0.398i)14-s + (0.203 − 0.979i)15-s + (0.460 − 0.887i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.269i)2-s + (0.460 + 0.887i)3-s + (0.854 − 0.519i)4-s + (−0.775 − 0.631i)5-s + (0.682 + 0.730i)6-s + (−0.990 + 0.136i)7-s + (0.682 − 0.730i)8-s + (−0.576 + 0.816i)9-s + (−0.917 − 0.398i)10-s + (−0.0682 + 0.997i)11-s + (0.854 + 0.519i)12-s + (−0.334 − 0.942i)13-s + (−0.917 + 0.398i)14-s + (0.203 − 0.979i)15-s + (0.460 − 0.887i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.998 + 0.0450i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.998 + 0.0450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.332652606 + 0.03004588099i\)
\(L(\frac12)\) \(\approx\) \(1.332652606 + 0.03004588099i\)
\(L(1)\) \(\approx\) \(1.508829270 + 0.01651739324i\)
\(L(1)\) \(\approx\) \(1.508829270 + 0.01651739324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (0.962 - 0.269i)T \)
3 \( 1 + (0.460 + 0.887i)T \)
5 \( 1 + (-0.775 - 0.631i)T \)
7 \( 1 + (-0.990 + 0.136i)T \)
11 \( 1 + (-0.0682 + 0.997i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (0.962 + 0.269i)T \)
29 \( 1 + (-0.334 + 0.942i)T \)
31 \( 1 + (0.460 - 0.887i)T \)
37 \( 1 + (-0.917 - 0.398i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (0.682 + 0.730i)T \)
59 \( 1 + (0.854 + 0.519i)T \)
61 \( 1 + (-0.917 + 0.398i)T \)
67 \( 1 + (-0.990 - 0.136i)T \)
71 \( 1 + (0.962 + 0.269i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (0.203 - 0.979i)T \)
83 \( 1 + (-0.0682 + 0.997i)T \)
89 \( 1 + (-0.775 - 0.631i)T \)
97 \( 1 + (0.460 + 0.887i)T \)
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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

−34.331401480298206522285034110108, −32.55295762564614517198420922203, −31.67955901502825625523980113235, −30.75236812832279165606709622754, −29.79560905694960031767692442395, −28.85720717554296284420273018574, −26.492987887284124325707376838414, −25.8944950377307120211986898789, −24.40858462817196605523425138644, −23.589149103799783921554801891620, −22.60267935322422441824366184767, −21.25802428581839521456980288727, −19.44486820136920854181898452990, −19.17443881921856392001111647151, −17.02581740758096439948964756791, −15.63388433280456610300882893762, −14.46789913255941297455664837696, −13.35227581968450979355176943331, −12.26343277198395745342144932314, −10.990117059566438455915844825172, −8.532243710653705659140507477553, −7.07001732369226753931987586746, −6.28921819437613647477591305682, −3.85522928736977886773070235763, −2.68281503458342409014602386626, 2.84522823346548694262016337874, 4.13510287445514101048688333155, 5.33364335886817658482282018786, 7.415154299485583695955339488146, 9.314750184413034882276520445227, 10.57512152753373067584237222065, 12.184278402602451252589133947933, 13.18328668058326132382760955070, 14.898530798193868926279632425325, 15.627532004008247588170059963, 16.66249672212835606397890693825, 19.26710005989781703626016338682, 20.151078233549935057544656299049, 20.947398829160435009805207230435, 22.531364628851368525446571963, 23.02931972222932863538701737117, 24.801566860244628917066566510224, 25.67653811882036916707541503147, 27.396207777761531001170396597655, 28.27533623632207343510955140176, 29.57626743920838919788031004563, 31.19606686141893109848988620392, 31.723798025332823509390065426797, 32.634311734363276741199722735577, 33.62738519590339069994778171519

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