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Properties

Label 1-47-47.4-r0-0-0
Degree $1$
Conductor $47$
Sign $0.999 + 0.0106i$
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  + (0.854 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (0.203 − 0.979i)5-s + (−0.0682 − 0.997i)6-s + (0.962 + 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (0.682 − 0.730i)10-s + (−0.990 + 0.136i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (0.682 + 0.730i)14-s + (−0.917 + 0.398i)15-s + (−0.576 + 0.816i)16-s + (−0.990 − 0.136i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (0.203 − 0.979i)5-s + (−0.0682 − 0.997i)6-s + (0.962 + 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (0.682 − 0.730i)10-s + (−0.990 + 0.136i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (0.682 + 0.730i)14-s + (−0.917 + 0.398i)15-s + (−0.576 + 0.816i)16-s + (−0.990 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.999 + 0.0106i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.999 + 0.0106i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140223806 + 0.006081840506i\)
\(L(\frac12)\) \(\approx\) \(1.140223806 + 0.006081840506i\)
\(L(1)\) \(\approx\) \(1.303363049 + 0.03364769467i\)
\(L(1)\) \(\approx\) \(1.303363049 + 0.03364769467i\)

Euler product

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

−33.75203799267550168436978077182, −33.163515529469768457576072482031, −31.66533549131130175712486080124, −30.70431759726963927451888855164, −29.45308966800928449943150809881, −28.58526581655189866722534752832, −27.19268271536565618568936170739, −26.22975371202241812857905745276, −24.21813639016928386382351806859, −23.30203234537203627262993084613, −22.0426699690331772015941421729, −21.4528279725256770772910945215, −20.2641218945892665658396662020, −18.6016690856820010853545553002, −17.29233954304948870057422571540, −15.47699777348451224208089547115, −14.74171540503330720002844377370, −13.39372166554518909674346360003, −11.428541422029054218580029306284, −10.96196548331125460975367172037, −9.66060634371771155443950025972, −7.03672789072035358618485347986, −5.456122360823906445734029553409, −4.29187784101055679712846227273, −2.54340442802492537267313239444, 2.15012713614538051693683078677, 4.8715051210256326887692536029, 5.56031322815730618082656727200, 7.381057369191659220880766679124, 8.41909820953279411933522577386, 11.03780151811351409738713025827, 12.41504167491316678148298117392, 13.05059617285777541857116398154, 14.498252235370696430993280135623, 16.03325152724628662437632069752, 17.224171836994701101721924529383, 18.10196477380953993606284754932, 20.162911911994182034269050434, 21.237309854806246324427762984804, 22.56596534572339700521551870284, 23.80131131212537353267197215060, 24.49103794744664959525120373881, 25.26297980959547280345660896454, 27.1484595604864606902503503101, 28.66522528116976210953953393005, 29.51839081808670239067539429990, 30.92388328212086009281460989057, 31.57339562951620355945541228012, 33.10550751482844421460591484601, 34.01605269632878925980675906295

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