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Properties

Label 1-3143-3143.3142-r1-0-0
Degree $1$
Conductor $3143$
Sign $1$
Analytic cond. $337.762$
Root an. cond. $337.762$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 29-s − 30-s + 31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 29-s − 30-s + 31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3143\)    =    \(7 \cdot 449\)
Sign: $1$
Analytic conductor: \(337.762\)
Root analytic conductor: \(337.762\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3143} (3142, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3143,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(9.150507428\)
\(L(\frac12)\) \(\approx\) \(9.150507428\)
\(L(1)\) \(\approx\) \(3.138092270\)
\(L(1)\) \(\approx\) \(3.138092270\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
449 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−18.86287235673773772396393733483, −18.61860898957282911412472615512, −17.09562158528776684795337338762, −16.435191068603374160037821262526, −15.74264739683401265293705962917, −15.14324150994159605663591620564, −14.69086938193490709123342510396, −13.782529672202559949022669080406, −13.50537240610034461200629176556, −12.402688017820324849765530645691, −11.997879423516267944342201596484, −11.23227088772130817759290547304, −10.45144652600284701104316040039, −9.49924994057834559238094079129, −8.630207391901468036260227417506, −7.980858140277461040721489616178, −7.17752070884935741707476347029, −6.733086848152657641700208813058, −5.60461415943744304876213551451, −4.729329498840697851433699583189, −3.871231744052237146538743412091, −3.45257941278261662645219363521, −2.89411505666425505159957771022, −1.557559912451354728844569517892, −1.03032608441514366408550900233, 1.03032608441514366408550900233, 1.557559912451354728844569517892, 2.89411505666425505159957771022, 3.45257941278261662645219363521, 3.871231744052237146538743412091, 4.729329498840697851433699583189, 5.60461415943744304876213551451, 6.733086848152657641700208813058, 7.17752070884935741707476347029, 7.980858140277461040721489616178, 8.630207391901468036260227417506, 9.49924994057834559238094079129, 10.45144652600284701104316040039, 11.23227088772130817759290547304, 11.997879423516267944342201596484, 12.402688017820324849765530645691, 13.50537240610034461200629176556, 13.782529672202559949022669080406, 14.69086938193490709123342510396, 15.14324150994159605663591620564, 15.74264739683401265293705962917, 16.435191068603374160037821262526, 17.09562158528776684795337338762, 18.61860898957282911412472615512, 18.86287235673773772396393733483

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