# Properties

 Label 2-2e8-1.1-c1-0-4 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 − 2·3-s + 9-s − 6·11-s − 6·17-s − 2·19-s − 5·25-s + 4·27-s + 12·33-s + 6·41-s + 10·43-s − 7·49-s + 12·51-s + 4·57-s − 6·59-s + 14·67-s − 2·73-s + 10·75-s − 11·81-s − 18·83-s − 18·89-s + 10·97-s − 6·99-s − 6·107-s + 18·113-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1/3·9-s − 1.80·11-s − 1.45·17-s − 0.458·19-s − 25-s + 0.769·27-s + 2.08·33-s + 0.937·41-s + 1.52·43-s − 49-s + 1.68·51-s + 0.529·57-s − 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.15·75-s − 1.22·81-s − 1.97·83-s − 1.90·89-s + 1.01·97-s − 0.603·99-s − 0.580·107-s + 1.69·113-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 + 2 T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 10 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 14 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 18 T + p T^{2}$$
89 $$1 + 18 T + p T^{2}$$
97 $$1 - 10 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$