# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 21-s + 6·23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 6·29-s − 30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$51870$$    =    $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{51870} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 51870,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.9616778313$$ $$L(\frac12)$$ $$\approx$$ $$0.9616778313$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;19\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;7,\;13,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
5 $$1 + T$$
7 $$1 - T$$
13 $$1 + T$$
19 $$1 + T$$
good11 $$1 + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + 6 T + p T^{2}$$
47 $$1 + 6 T + p T^{2}$$
53 $$1 + 12 T + p T^{2}$$
59 $$1 + 8 T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + 12 T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 - 6 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}