Properties

Degree 2
Conductor 7
Sign $1$
Motivic weight 8
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 31·2-s + 705·4-s + 2.40e3·7-s − 1.39e4·8-s + 6.56e3·9-s + 1.31e4·11-s − 7.44e4·14-s + 2.51e5·16-s − 2.03e5·18-s − 4.07e5·22-s − 2.09e4·23-s + 3.90e5·25-s + 1.69e6·28-s + 1.08e5·29-s − 4.21e6·32-s + 4.62e6·36-s − 2.07e6·37-s − 6.72e6·43-s + 9.27e6·44-s + 6.48e5·46-s + 5.76e6·49-s − 1.21e7·50-s + 1.53e7·53-s − 3.34e7·56-s − 3.35e6·58-s + 1.57e7·63-s + 6.65e7·64-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.75·4-s + 7-s − 3.39·8-s + 9-s + 0.898·11-s − 1.93·14-s + 3.83·16-s − 1.93·18-s − 1.74·22-s − 0.0747·23-s + 25-s + 2.75·28-s + 0.152·29-s − 4.02·32-s + 2.75·36-s − 1.10·37-s − 1.96·43-s + 2.47·44-s + 0.144·46-s + 49-s − 1.93·50-s + 1.94·53-s − 3.39·56-s − 0.296·58-s + 63-s + 3.96·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(8\)
character  :  $\chi_{7} (6, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7,\ (\ :4),\ 1)$
$L(\frac{9}{2})$  $\approx$  $0.705076$
$L(\frac12)$  $\approx$  $0.705076$
$L(5)$   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 \( 1 - p^{4} T \)
good2 \( 1 + 31 T + p^{8} T^{2} \)
3 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
5 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
11 \( 1 - 13154 T + p^{8} T^{2} \)
13 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
17 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
19 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
23 \( 1 + 20926 T + p^{8} T^{2} \)
29 \( 1 - 108194 T + p^{8} T^{2} \)
31 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
37 \( 1 + 2073886 T + p^{8} T^{2} \)
41 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
43 \( 1 + 6726046 T + p^{8} T^{2} \)
47 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
53 \( 1 - 15377762 T + p^{8} T^{2} \)
59 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
61 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
67 \( 1 + 15839326 T + p^{8} T^{2} \)
71 \( 1 + 42331966 T + p^{8} T^{2} \)
73 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
79 \( 1 + 64606846 T + p^{8} T^{2} \)
83 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
89 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
97 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
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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

−20.19919249791923172639004889380, −18.83057030527403870855414681003, −17.78984690345478177936540549681, −16.56263945714005997042348548484, −15.04192493919052488382018448312, −11.80658495895418354406187242632, −10.31503820493995659736389591099, −8.692640034198523192534860102525, −7.06661744652306668936091890853, −1.43975694701657115285982645837, 1.43975694701657115285982645837, 7.06661744652306668936091890853, 8.692640034198523192534860102525, 10.31503820493995659736389591099, 11.80658495895418354406187242632, 15.04192493919052488382018448312, 16.56263945714005997042348548484, 17.78984690345478177936540549681, 18.83057030527403870855414681003, 20.19919249791923172639004889380

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