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Properties

Label 1-47-47.36-r0-0-0
Degree $1$
Conductor $47$
Sign $0.204 - 0.978i$
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  + (−0.0682 − 0.997i)2-s + (0.962 + 0.269i)3-s + (−0.990 + 0.136i)4-s + (−0.576 − 0.816i)5-s + (0.203 − 0.979i)6-s + (0.682 − 0.730i)7-s + (0.203 + 0.979i)8-s + (0.854 + 0.519i)9-s + (−0.775 + 0.631i)10-s + (−0.917 − 0.398i)11-s + (−0.990 − 0.136i)12-s + (0.460 + 0.887i)13-s + (−0.775 − 0.631i)14-s + (−0.334 − 0.942i)15-s + (0.962 − 0.269i)16-s + (−0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (−0.0682 − 0.997i)2-s + (0.962 + 0.269i)3-s + (−0.990 + 0.136i)4-s + (−0.576 − 0.816i)5-s + (0.203 − 0.979i)6-s + (0.682 − 0.730i)7-s + (0.203 + 0.979i)8-s + (0.854 + 0.519i)9-s + (−0.775 + 0.631i)10-s + (−0.917 − 0.398i)11-s + (−0.990 − 0.136i)12-s + (0.460 + 0.887i)13-s + (−0.775 − 0.631i)14-s + (−0.334 − 0.942i)15-s + (0.962 − 0.269i)16-s + (−0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.204 - 0.978i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7272400807 - 0.5908422267i\)
\(L(\frac12)\) \(\approx\) \(0.7272400807 - 0.5908422267i\)
\(L(1)\) \(\approx\) \(0.9485491760 - 0.5242772741i\)
\(L(1)\) \(\approx\) \(0.9485491760 - 0.5242772741i\)

Euler product

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

−34.4571745934514785697490409980, −33.29037060638769145535309335309, −31.891489801121442590945340212910, −31.070332965384978955116533676785, −30.32372624307724955538441894690, −28.141046305341337099126245964861, −26.948505598356160233245696261860, −26.1061493709893727977481998185, −25.08112876869165878236362860920, −24.04631225098346605466243674319, −22.886159944714546331304967540739, −21.468055141630130163971381786715, −19.858672919404962069965488974756, −18.42123342780565365266391686546, −17.99078080593598640380420371860, −15.56263632659343248845824387175, −15.22738615909632986870256044015, −13.98230711342765938144649380436, −12.64848767291126717896202925094, −10.54297367934853169751540688712, −8.73894920761608930483067353826, −7.88828693476623610531340138875, −6.65057938498428846802571109799, −4.669702063065736122859407638238, −2.77315928330910291139521107398, 1.76763548524197040750084256516, 3.74835035299101187462966892948, 4.69839738432210755448877399880, 7.94111702584777097822443037368, 8.69765368269078292975707988439, 10.22128985377053295156521724045, 11.50490800566421636668029904994, 13.126453222758373207754702054553, 13.93870132453826206844161087356, 15.586255533101081971759976829433, 17.11579587623219228900852828122, 18.79788718754665375558278994040, 19.75551456066625411230498978686, 20.82575332488544003493703580528, 21.323848162642478976726269317543, 23.327281102165559295703350884300, 24.27287363492624432440354253649, 26.154573073413784461372072660554, 26.97829281045485658505865339000, 27.961884959011884371382949932222, 29.24716762919445831333444669298, 30.65873341958639400888060517052, 31.33410341669606525864022376535, 32.23145608160948991093149534235, 33.52347477689713616931130603392

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