# Properties

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

# Origins

## 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 )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}