# Properties

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

# Origins

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