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Properties

Label 1-47-47.42-r0-0-0
Degree $1$
Conductor $47$
Sign $0.0256 - 0.999i$
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  + (−0.775 + 0.631i)2-s + (−0.917 + 0.398i)3-s + (0.203 − 0.979i)4-s + (−0.990 − 0.136i)5-s + (0.460 − 0.887i)6-s + (−0.334 − 0.942i)7-s + (0.460 + 0.887i)8-s + (0.682 − 0.730i)9-s + (0.854 − 0.519i)10-s + (−0.576 − 0.816i)11-s + (0.203 + 0.979i)12-s + (−0.0682 − 0.997i)13-s + (0.854 + 0.519i)14-s + (0.962 − 0.269i)15-s + (−0.917 − 0.398i)16-s + (−0.576 + 0.816i)17-s + ⋯
L(s)  = 1  + (−0.775 + 0.631i)2-s + (−0.917 + 0.398i)3-s + (0.203 − 0.979i)4-s + (−0.990 − 0.136i)5-s + (0.460 − 0.887i)6-s + (−0.334 − 0.942i)7-s + (0.460 + 0.887i)8-s + (0.682 − 0.730i)9-s + (0.854 − 0.519i)10-s + (−0.576 − 0.816i)11-s + (0.203 + 0.979i)12-s + (−0.0682 − 0.997i)13-s + (0.854 + 0.519i)14-s + (0.962 − 0.269i)15-s + (−0.917 − 0.398i)16-s + (−0.576 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0256 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0256 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.0256 - 0.999i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.0256 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1429997235 - 0.1393833135i\)
\(L(\frac12)\) \(\approx\) \(0.1429997235 - 0.1393833135i\)
\(L(1)\) \(\approx\) \(0.3670962563 + 0.004963282090i\)
\(L(1)\) \(\approx\) \(0.3670962563 + 0.004963282090i\)

Euler product

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

−34.46333179583129068806909660100, −33.91324636482783967872926126944, −31.54652429648775521720129957028, −30.79748350391039606858864135016, −29.4750843441775350940567014743, −28.40294919977373612215496760781, −27.83634038194739152961872605690, −26.52201228950834197373263562932, −25.16238593979568723612818242219, −23.69882825338759177731635242611, −22.50915381021894758013560122680, −21.42255041718053287761563001547, −19.75529623556168069202485589393, −18.75793676457045366784732458905, −17.951588962598330259958988534056, −16.45531843013995263693573036126, −15.50932302878179891782942166196, −12.91566662692489963615956598742, −11.94145361766409655690301237380, −11.1396072972401157455670011483, −9.53896537682425020069526498616, −7.901678991042837711866479139041, −6.66572142929019726757433147837, −4.452266836931840609780563149634, −2.270113657768369340244224875159, 0.38349156876872986168624001128, 4.088346918029371312751854393953, 5.75364005603345942945235797973, 7.16877186791161224890787434124, 8.49291319040452015767295674527, 10.40774485219283101375474196290, 10.9728297182213592850301839188, 12.82969274558451488662836541779, 14.91133930078839336855920976562, 16.040377347708285565583481871614, 16.74706205753009582738151288348, 17.99848898872060177912581145383, 19.34416142152110587386075135040, 20.4723141828270918662984740787, 22.40346397733643490904329337870, 23.57410915856156748033915110596, 24.06567167223093954920503526309, 26.025994937687829193214180023154, 26.99727223522711654824605187642, 27.71256900286323288504981396544, 28.8813353222821154536201903500, 29.966185785781249583428944649246, 32.05190729192350912467077998796, 32.80333991472905561294161538671, 34.09901428537311476025169551096

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