Properties

Degree $2$
Conductor $7$
Sign $1$
Motivic weight $2$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 5·4-s − 7·7-s − 3·8-s + 9·9-s − 6·11-s + 21·14-s − 11·16-s − 27·18-s + 18·22-s + 18·23-s + 25·25-s − 35·28-s − 54·29-s + 45·32-s + 45·36-s − 38·37-s + 58·43-s − 30·44-s − 54·46-s + 49·49-s − 75·50-s − 6·53-s + 21·56-s + 162·58-s − 63·63-s − 91·64-s + ⋯
L(s)  = 1  − 3/2·2-s + 5/4·4-s − 7-s − 3/8·8-s + 9-s − 0.545·11-s + 3/2·14-s − 0.687·16-s − 3/2·18-s + 9/11·22-s + 0.782·23-s + 25-s − 5/4·28-s − 1.86·29-s + 1.40·32-s + 5/4·36-s − 1.02·37-s + 1.34·43-s − 0.681·44-s − 1.17·46-s + 49-s − 3/2·50-s − 0.113·53-s + 3/8·56-s + 2.79·58-s − 63-s − 1.42·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Motivic weight: \(2\)
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.332981\)
\(L(\frac12)\) \(\approx\) \(0.332981\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + p T \)
good2 \( 1 + 3 T + p^{2} T^{2} \)
3 \( ( 1 - p T )( 1 + p T ) \)
5 \( ( 1 - p T )( 1 + p T ) \)
11 \( 1 + 6 T + p^{2} T^{2} \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 - 18 T + p^{2} T^{2} \)
29 \( 1 + 54 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 + 38 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 58 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 + 6 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 + 118 T + p^{2} T^{2} \)
71 \( 1 - 114 T + p^{2} T^{2} \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 + 94 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−22.54364926152355667236424777146, −20.72321683128359948995825436530, −19.21391450210761761408870427785, −18.40497630679973574791649581702, −16.79600315524904172443777154084, −15.64810467259429164162308409825, −12.96585968300816620901605936259, −10.58097485411185119630259735576, −9.256810748581498587619513382329, −7.21458918128718444354242474222, 7.21458918128718444354242474222, 9.256810748581498587619513382329, 10.58097485411185119630259735576, 12.96585968300816620901605936259, 15.64810467259429164162308409825, 16.79600315524904172443777154084, 18.40497630679973574791649581702, 19.21391450210761761408870427785, 20.72321683128359948995825436530, 22.54364926152355667236424777146

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