Properties

Label 1-47-47.21-r0-0-0
Degree $1$
Conductor $47$
Sign $0.528 + 0.849i$
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  + (−0.576 + 0.816i)2-s + (−0.775 − 0.631i)3-s + (−0.334 − 0.942i)4-s + (0.682 + 0.730i)5-s + (0.962 − 0.269i)6-s + (0.460 + 0.887i)7-s + (0.962 + 0.269i)8-s + (0.203 + 0.979i)9-s + (−0.990 + 0.136i)10-s + (0.854 − 0.519i)11-s + (−0.334 + 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.990 − 0.136i)14-s + (−0.0682 − 0.997i)15-s + (−0.775 + 0.631i)16-s + (0.854 + 0.519i)17-s + ⋯
L(s)  = 1  + (−0.576 + 0.816i)2-s + (−0.775 − 0.631i)3-s + (−0.334 − 0.942i)4-s + (0.682 + 0.730i)5-s + (0.962 − 0.269i)6-s + (0.460 + 0.887i)7-s + (0.962 + 0.269i)8-s + (0.203 + 0.979i)9-s + (−0.990 + 0.136i)10-s + (0.854 − 0.519i)11-s + (−0.334 + 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.990 − 0.136i)14-s + (−0.0682 − 0.997i)15-s + (−0.775 + 0.631i)16-s + (0.854 + 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.528 + 0.849i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.528 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5097529614 + 0.2831814973i\)
\(L(\frac12)\) \(\approx\) \(0.5097529614 + 0.2831814973i\)
\(L(1)\) \(\approx\) \(0.6620541170 + 0.2379608472i\)
\(L(1)\) \(\approx\) \(0.6620541170 + 0.2379608472i\)

Euler product

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

−33.83776598970213574095796211320, −32.86534020044981668772451247520, −31.62245140285900050509515875570, −29.73791068738570674157982328677, −29.4317412824460609337098960440, −27.877520270718032969339867570889, −27.46820670273517141577271665582, −26.12430275883080153771127710837, −24.6269030946894436673591427484, −22.95004971057606116037913138758, −21.87725537502151798664210006280, −20.74603837294657992881284372007, −20.02265932495824466732015870337, −18.01566687983787738465933223124, −17.14748561164939897368929943428, −16.46917898147855990782428326406, −14.26327192979446438137165591040, −12.60396666667644789691443936281, −11.568080442563174648849166443649, −10.08129300318320521464005349378, −9.45008254812662652370149508025, −7.48530408255412850058595489129, −5.2527132513863644535270063191, −3.91585638446048628515317885624, −1.3409041402870911383977601184, 1.88162549773810798823313482313, 5.294408712182515313868395516551, 6.27327136731265444510281126551, 7.462628094201174899106110085567, 9.14364379609625543454810997120, 10.62479143944238966614282521334, 11.98126771672208577441454896397, 13.874810114550153733317368731583, 14.85589189212300209570503667190, 16.56349635074765193056837362824, 17.53706515731354540336277989715, 18.46653157630975510094824688097, 19.32416823719637945855113584490, 21.79057480445477267887248195809, 22.55839784603781435473827822728, 24.17780208368806602066944254461, 24.74527190158246894360693207972, 25.98886820155196004382243069418, 27.35032241138250778597553759335, 28.39295923222660297651250459042, 29.44535430309857954512877299570, 30.64066847064551582461905118469, 32.3256885816564347191111858894, 33.56480443107222262432685945389, 34.50440171525018292637002246979

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