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Properties

Label 2-3143-3143.3142-c0-0-4
Degree $2$
Conductor $3143$
Sign $1$
Analytic cond. $1.56856$
Root an. cond. $1.25242$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 0.801·6-s + 7-s − 2.24·8-s − 0.801·9-s + 1.24·11-s − 12-s − 1.80·13-s − 1.80·14-s + 1.80·16-s − 1.80·17-s + 1.44·18-s − 0.445·19-s − 0.445·21-s − 2.24·22-s − 0.445·23-s + 1.00·24-s + 25-s + 3.24·26-s + 0.801·27-s + 2.24·28-s + 2·31-s − 1.00·32-s − 0.554·33-s + 3.24·34-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 0.801·6-s + 7-s − 2.24·8-s − 0.801·9-s + 1.24·11-s − 12-s − 1.80·13-s − 1.80·14-s + 1.80·16-s − 1.80·17-s + 1.44·18-s − 0.445·19-s − 0.445·21-s − 2.24·22-s − 0.445·23-s + 1.00·24-s + 25-s + 3.24·26-s + 0.801·27-s + 2.24·28-s + 2·31-s − 1.00·32-s − 0.554·33-s + 3.24·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3143\)    =    \(7 \cdot 449\)
Sign: $1$
Analytic conductor: \(1.56856\)
Root analytic conductor: \(1.25242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3143} (3142, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3143,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4219232694\)
\(L(\frac12)\) \(\approx\) \(0.4219232694\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
449 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
3 \( 1 + 0.445T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 + 1.80T + T^{2} \)
17 \( 1 + 1.80T + T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + 0.445T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.24T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.879972598005001093954043727840, −8.297039851744602462819813675575, −7.58772560060690902325322448759, −6.65334065656875661580820056701, −6.41317205529543759548951519383, −5.06511196790240138349988853019, −4.35832513724921765585448668536, −2.62531106396894210903894558673, −2.07059625139617855990749143752, −0.75765433953093031730356152971, 0.75765433953093031730356152971, 2.07059625139617855990749143752, 2.62531106396894210903894558673, 4.35832513724921765585448668536, 5.06511196790240138349988853019, 6.41317205529543759548951519383, 6.65334065656875661580820056701, 7.58772560060690902325322448759, 8.297039851744602462819813675575, 8.879972598005001093954043727840

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