# Properties

 Degree 2 Conductor $2 \cdot 5 \cdot 11 \cdot 43$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s + 28-s − 5·29-s − 3·30-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 0.188·28-s − 0.928·29-s − 0.547·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4730$$    =    $$2 \cdot 5 \cdot 11 \cdot 43$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{4730} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 4730,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;11,\;43\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
5 $$1 - T$$
11 $$1 - T$$
43 $$1 - T$$
good3 $$1 + p T + p T^{2}$$
7 $$1 - T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 3 T + p T^{2}$$
19 $$1 + T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 5 T + p T^{2}$$
31 $$1 + 3 T + p T^{2}$$
37 $$1 + 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 9 T + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 - 13 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 - 9 T + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 + 4 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + T + p T^{2}$$
97 $$1 + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}