# Properties

 Label 2-6930-1.1-c1-0-6 Degree $2$ Conductor $6930$ Sign $1$ Analytic cond. $55.3363$ Root an. cond. $7.43883$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s − 4·13-s − 14-s + 16-s + 2·17-s − 6·19-s − 20-s + 22-s + 6·23-s + 25-s + 4·26-s + 28-s − 2·29-s − 32-s − 2·34-s − 35-s − 10·37-s + 6·38-s + 40-s + 6·41-s − 2·43-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 1.37·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.342·34-s − 0.169·35-s − 1.64·37-s + 0.973·38-s + 0.158·40-s + 0.937·41-s − 0.304·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6930$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Sign: $1$ Analytic conductor: $$55.3363$$ Root analytic conductor: $$7.43883$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 6930,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9167008376$$ $$L(\frac12)$$ $$\approx$$ $$0.9167008376$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1$$
5 $$1 + T$$
7 $$1 - T$$
11 $$1 + T$$
good13 $$1 + 4 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 - 2 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 4 T + p T^{2}$$
67 $$1 + 8 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 4 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$