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Properties

Label 1-47-47.15-r1-0-0
Degree $1$
Conductor $47$
Sign $-0.978 - 0.204i$
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  + (0.203 + 0.979i)2-s + (0.682 + 0.730i)3-s + (−0.917 + 0.398i)4-s + (−0.962 + 0.269i)5-s + (−0.576 + 0.816i)6-s + (−0.775 − 0.631i)7-s + (−0.576 − 0.816i)8-s + (−0.0682 + 0.997i)9-s + (−0.460 − 0.887i)10-s + (0.334 + 0.942i)11-s + (−0.917 − 0.398i)12-s + (0.990 + 0.136i)13-s + (0.460 − 0.887i)14-s + (−0.854 − 0.519i)15-s + (0.682 − 0.730i)16-s + (−0.334 + 0.942i)17-s + ⋯
L(s)  = 1  + (0.203 + 0.979i)2-s + (0.682 + 0.730i)3-s + (−0.917 + 0.398i)4-s + (−0.962 + 0.269i)5-s + (−0.576 + 0.816i)6-s + (−0.775 − 0.631i)7-s + (−0.576 − 0.816i)8-s + (−0.0682 + 0.997i)9-s + (−0.460 − 0.887i)10-s + (0.334 + 0.942i)11-s + (−0.917 − 0.398i)12-s + (0.990 + 0.136i)13-s + (0.460 − 0.887i)14-s + (−0.854 − 0.519i)15-s + (0.682 − 0.730i)16-s + (−0.334 + 0.942i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1218918698 + 1.177345140i\)
\(L(\frac12)\) \(\approx\) \(-0.1218918698 + 1.177345140i\)
\(L(1)\) \(\approx\) \(0.5929139897 + 0.8216356878i\)
\(L(1)\) \(\approx\) \(0.5929139897 + 0.8216356878i\)

Euler product

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

−32.453500046698524723431819988117, −31.7552984156854697395576287821, −30.92407042799872480787925724682, −29.835290124272204021693204195627, −28.77683445125979005625949284990, −27.55802888019914414296169606514, −26.36372139084297399958456996824, −24.85928306852880625717650494464, −23.62715880114517675135870362493, −22.61765256987173577456703877512, −21.06018370072360104685829009330, −19.904477074876565843057365577292, −19.083773924939459446499773438240, −18.295062238402518624282078636819, −16.09184911646469929073034793904, −14.59647715831398320443487677822, −13.236417048587677347913225940119, −12.3415492911283555107103939971, −11.16762394147771653484884063701, −9.138176823569226918962954940054, −8.31544950175962087695848685029, −6.21392939354554593651953220302, −3.93350257849576094131322227614, −2.730409298073386134600771707111, −0.63175855690870534422657142295, 3.57066013974566632707521118042, 4.37224819235509026690858213360, 6.58519134342189345706720739670, 7.88540628164554380906819846103, 9.12997071731145234213078216048, 10.60707381930576907695617603196, 12.729916263610924775649633309089, 14.05282560036798041296920439715, 15.29043823767866198708296280750, 15.92150281776180142334785438955, 17.22833217260014598473119726338, 19.06735484711205059562441487935, 20.065412369473588484110420002840, 21.69373230215688498707930341918, 22.89750521464894718730332866427, 23.68943425880625015540901814775, 25.542413486337256202512907336064, 26.00848764098621803456180249039, 27.178971299594985910373314973173, 28.06419942774514859997322080934, 30.45410604492584402286258766550, 31.12663253351899657683323332160, 32.40125718505968667947917447355, 33.01547074077708643539069559159, 34.23683780182612125873142331611

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