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Properties

Label 1-47-47.25-r0-0-0
Degree $1$
Conductor $47$
Sign $0.0296 - 0.999i$
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  + (0.203 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0296 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0296 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.0296 - 0.999i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.0296 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7325044318 - 0.7545402156i\)
\(L(\frac12)\) \(\approx\) \(0.7325044318 - 0.7545402156i\)
\(L(1)\) \(\approx\) \(0.9893038110 - 0.6886998446i\)
\(L(1)\) \(\approx\) \(0.9893038110 - 0.6886998446i\)

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

−34.04957098102760512974637187167, −32.94828456298864398013433836658, −32.41486194658608838642479399057, −31.47568522206174043310663005148, −29.899267190286780232493987189695, −28.45948949062004646309248533353, −26.72571511064285398618204776775, −26.33722882709472364846560917005, −25.11398898854940868806952511172, −24.18926579462292491506677288442, −22.40642057422020774626716968132, −21.70332958755505131005627804930, −20.325846195869297212361715008254, −18.791447052081855829105995807888, −17.01608421203720469325650018560, −16.397929823716583356211459158608, −14.9667618860153719008355973007, −13.80740320356456918336583421324, −13.01684532205419788440663595133, −10.24289215909095701261668372315, −9.31713894159671165257166745145, −7.95873813543737516415868282310, −6.21215442626191269096894183082, −4.789490575877882376781363850471, −3.16040985831181206234352890497, 2.037871138241561321859580388892, 3.009089610859768637736380604133, 5.34336438591355465742472792657, 7.09639844959235201723596721483, 9.23203921626474001137684469618, 9.81769853974332040826247935876, 11.90040397358835589585852564789, 12.96419177094793418825781075965, 13.8842423695235674801672167289, 15.13723311322744521997811697469, 17.62291029696487199482675594790, 18.43718312566399927034708619692, 19.5751829008931026140233408455, 20.62368426033586906915751669183, 21.89428695624010385583698880274, 22.91078394162421931993144752922, 24.55060868042220806503636919673, 25.62166765972686913118323349086, 26.73356969124632104687977805511, 28.621752182847639946073646656232, 29.206790735810270618298165351547, 30.24966087973127795631631125304, 31.38834473164760024912194758338, 32.14728576553807559558489794633, 33.5873961939908612141487535475

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