# Properties

 Label 2-2e8-1.1-c1-0-5 Degree $2$ Conductor $256$ Sign $-1$ Analytic cond. $2.04417$ Root an. cond. $1.42974$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4·5-s − 3·9-s − 4·13-s − 2·17-s + 11·25-s − 4·29-s + 12·37-s − 10·41-s + 12·45-s − 7·49-s − 4·53-s + 12·61-s + 16·65-s − 6·73-s + 9·81-s + 8·85-s + 10·89-s − 18·97-s − 20·101-s − 20·109-s − 14·113-s + 12·117-s + ⋯
 L(s)  = 1 − 1.78·5-s − 9-s − 1.10·13-s − 0.485·17-s + 11/5·25-s − 0.742·29-s + 1.97·37-s − 1.56·41-s + 1.78·45-s − 49-s − 0.549·53-s + 1.53·61-s + 1.98·65-s − 0.702·73-s + 81-s + 0.867·85-s + 1.05·89-s − 1.82·97-s − 1.99·101-s − 1.91·109-s − 1.31·113-s + 1.10·117-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$256$$    =    $$2^{8}$$ Sign: $-1$ Analytic conductor: $$2.04417$$ Root analytic conductor: $$1.42974$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{256} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 256,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + p T^{2}$$
5 $$1 + 4 T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 4 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 12 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 4 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 12 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 + 18 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$