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Properties

Label 1-47-47.20-r1-0-0
Degree $1$
Conductor $47$
Sign $0.820 - 0.571i$
Analytic cond. $5.05085$
Root an. cond. $5.05085$
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  + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (0.334 − 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (−0.203 + 0.979i)10-s + (−0.682 − 0.730i)11-s + (0.962 + 0.269i)12-s + (0.576 − 0.816i)13-s + (0.203 + 0.979i)14-s + (0.775 − 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (0.334 − 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (−0.203 + 0.979i)10-s + (−0.682 − 0.730i)11-s + (0.962 + 0.269i)12-s + (0.576 − 0.816i)13-s + (0.203 + 0.979i)14-s + (0.775 − 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.820 - 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279188602 - 0.4016742898i\)
\(L(\frac12)\) \(\approx\) \(1.279188602 - 0.4016742898i\)
\(L(1)\) \(\approx\) \(1.002771593 - 0.1151531257i\)
\(L(1)\) \(\approx\) \(1.002771593 - 0.1151531257i\)

Euler product

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

−34.300555149842250724846337587765, −32.95085917705153635370297413652, −31.08128735813187088365377194459, −30.53031448733966003639112243146, −29.19994961165131746731692904705, −28.208333211242565954131070920615, −26.58604286275147722182037725559, −25.81283299234629726707449885790, −25.13480504570999481082058889713, −23.66982701781179936105182370652, −21.68329468064430656293042493782, −20.71530059160087084260235100789, −19.14457193608284990693418860795, −18.6144021148156991519223694030, −17.60228745981598182343908438454, −15.60163659648015711272648191287, −14.73197632628388633713099410428, −12.99164883725680320400885054950, −11.53528032982597841787698763278, −9.9478422952518535068443911003, −8.84823980011426108832625507121, −7.485133535313558420739334174642, −6.30610188220571258220618407645, −3.02480223128325273103461001964, −1.94099151002913458754789970594, 1.097785436578513746532622581664, 3.23932574000450466885325527154, 5.37723064831182448964399661650, 7.58307835761139320581199331082, 8.56703439141542666967511699670, 9.805307802047656752873457769318, 10.814031843689237664835140934552, 12.98793495445818246159460535768, 14.31138974336861170180708762803, 16.03876627439184325223221989152, 16.53883194542810459876262847849, 18.12702500010543691397955929969, 19.56865991372163910558288713337, 20.558628095059880637860501085346, 21.13506693391528257260411496135, 23.44633509371685328628190625941, 24.83313414664997407696208767750, 25.57542348958479806964213798395, 26.855495481127432601340728280859, 27.51568807019945729079307427890, 28.944369152803292602748190256021, 29.94115816453180380832136743642, 31.61703345311415403089270445184, 32.80102266118270362624818692820, 33.42876377138129973386208747904

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