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

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{7} (6, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7,\ (\ :1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.332981$
$L(\frac12)$  $\approx$  $0.332981$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \(F_p\) is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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