Properties

Label 1-47-47.6-r0-0-0
Degree $1$
Conductor $47$
Sign $-0.216 - 0.976i$
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  + (0.682 − 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (0.460 − 0.887i)5-s + (−0.775 + 0.631i)6-s + (−0.917 + 0.398i)7-s + (−0.775 − 0.631i)8-s + (0.962 + 0.269i)9-s + (−0.334 − 0.942i)10-s + (0.203 − 0.979i)11-s + (−0.0682 + 0.997i)12-s + (0.854 + 0.519i)13-s + (−0.334 + 0.942i)14-s + (−0.576 + 0.816i)15-s + (−0.990 + 0.136i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (0.460 − 0.887i)5-s + (−0.775 + 0.631i)6-s + (−0.917 + 0.398i)7-s + (−0.775 − 0.631i)8-s + (0.962 + 0.269i)9-s + (−0.334 − 0.942i)10-s + (0.203 − 0.979i)11-s + (−0.0682 + 0.997i)12-s + (0.854 + 0.519i)13-s + (−0.334 + 0.942i)14-s + (−0.576 + 0.816i)15-s + (−0.990 + 0.136i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.216 - 0.976i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ -0.216 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5553186373 - 0.6922955192i\)
\(L(\frac12)\) \(\approx\) \(0.5553186373 - 0.6922955192i\)
\(L(1)\) \(\approx\) \(0.8521380698 - 0.6029344270i\)
\(L(1)\) \(\approx\) \(0.8521380698 - 0.6029344270i\)

Euler product

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

−34.29671842361156886429131898800, −33.01477824435120702815791422167, −32.82484785429606973964655821975, −30.86314017766625021514548729646, −29.906453144358137527751367763561, −28.89532556152456641055616081768, −27.25295858523637522589495379214, −26.0415665723696499925626427420, −25.137625035228278775749521448535, −23.449966477446293738453313145492, −22.69890221168015474837038744220, −22.08204158582698933344167404720, −20.57419313007232518298165158707, −18.38260018830316973780441231899, −17.47349477033308828273193042325, −16.23020777667292100007002872234, −15.210182583805480163869958694072, −13.6655071160404353666647823325, −12.54697512164292262280183689119, −11.0409575347891872839907802567, −9.62272972547230318186759990326, −7.11217113408887885864435729934, −6.45578303274011920408611017712, −4.9862060250003368995194729585, −3.25269961680124176012989509065, 1.39327769693533072537278646753, 3.77538198522237045906765133437, 5.525488674246419213121380325526, 6.232345595963924075414471595768, 9.051238582765079435594297005601, 10.39141968484930787892128555695, 11.77001603871096209284073446109, 12.76427413388063078559772608479, 13.701159610963848804430564250135, 15.7850890396662261894181991226, 16.76024780338913363822290360128, 18.44681045724713765171280204894, 19.47129535803374474452847394519, 21.1627058062742315324914589397, 21.81635944983139879853603649138, 23.105872027226819384884727455768, 24.031094187138494082233199922937, 25.196798984796508397923085519675, 27.322019423366439387584478494773, 28.60113144245468842696545380080, 28.90854140108378814804930996801, 30.03192723284680813800118127133, 31.54307966120165461009163791917, 32.623599915141208751680190925924, 33.32709559320414224341519085143

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