# Properties

 Label 2-2e3-1.1-c3-0-0 Degree $2$ Conductor $8$ Sign $1$ Analytic cond. $0.472015$ Root an. cond. $0.687033$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·3-s − 2·5-s + 24·7-s − 11·9-s − 44·11-s + 22·13-s + 8·15-s + 50·17-s + 44·19-s − 96·21-s − 56·23-s − 121·25-s + 152·27-s + 198·29-s − 160·31-s + 176·33-s − 48·35-s − 162·37-s − 88·39-s − 198·41-s + 52·43-s + 22·45-s + 528·47-s + 233·49-s − 200·51-s − 242·53-s + 88·55-s + ⋯
 L(s)  = 1 − 0.769·3-s − 0.178·5-s + 1.29·7-s − 0.407·9-s − 1.20·11-s + 0.469·13-s + 0.137·15-s + 0.713·17-s + 0.531·19-s − 0.997·21-s − 0.507·23-s − 0.967·25-s + 1.08·27-s + 1.26·29-s − 0.926·31-s + 0.928·33-s − 0.231·35-s − 0.719·37-s − 0.361·39-s − 0.754·41-s + 0.184·43-s + 0.0728·45-s + 1.63·47-s + 0.679·49-s − 0.549·51-s − 0.627·53-s + 0.215·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8$$    =    $$2^{3}$$ Sign: $1$ Analytic conductor: $$0.472015$$ Root analytic conductor: $$0.687033$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{8} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 8,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6900311631$$ $$L(\frac12)$$ $$\approx$$ $$0.6900311631$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 4 T + p^{3} T^{2}$$
5 $$1 + 2 T + p^{3} T^{2}$$
7 $$1 - 24 T + p^{3} T^{2}$$
11 $$1 + 4 p T + p^{3} T^{2}$$
13 $$1 - 22 T + p^{3} T^{2}$$
17 $$1 - 50 T + p^{3} T^{2}$$
19 $$1 - 44 T + p^{3} T^{2}$$
23 $$1 + 56 T + p^{3} T^{2}$$
29 $$1 - 198 T + p^{3} T^{2}$$
31 $$1 + 160 T + p^{3} T^{2}$$
37 $$1 + 162 T + p^{3} T^{2}$$
41 $$1 + 198 T + p^{3} T^{2}$$
43 $$1 - 52 T + p^{3} T^{2}$$
47 $$1 - 528 T + p^{3} T^{2}$$
53 $$1 + 242 T + p^{3} T^{2}$$
59 $$1 + 668 T + p^{3} T^{2}$$
61 $$1 - 550 T + p^{3} T^{2}$$
67 $$1 - 188 T + p^{3} T^{2}$$
71 $$1 - 728 T + p^{3} T^{2}$$
73 $$1 - 154 T + p^{3} T^{2}$$
79 $$1 + 656 T + p^{3} T^{2}$$
83 $$1 - 236 T + p^{3} T^{2}$$
89 $$1 - 714 T + p^{3} T^{2}$$
97 $$1 + 478 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$