# Properties

 Degree $2$ Conductor $128$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯
 L(s)  = 1 − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{128} (63, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5827916324$$ $$L(\frac12)$$ $$\approx$$ $$0.5827916324$$ $$L(1)$$ 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 + T^{2}$$
5 $$( 1 - T )( 1 + T )$$
7 $$( 1 - T )( 1 + T )$$
11 $$1 + T^{2}$$
13 $$( 1 - T )( 1 + T )$$
17 $$( 1 + T )^{2}$$
19 $$1 + T^{2}$$
23 $$( 1 - T )( 1 + T )$$
29 $$( 1 - T )( 1 + T )$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )( 1 + T )$$
41 $$( 1 - T )^{2}$$
43 $$1 + T^{2}$$
47 $$( 1 - T )( 1 + T )$$
53 $$( 1 - T )( 1 + T )$$
59 $$1 + T^{2}$$
61 $$( 1 - T )( 1 + T )$$
67 $$1 + T^{2}$$
71 $$( 1 - T )( 1 + T )$$
73 $$( 1 + T )^{2}$$
79 $$( 1 - T )( 1 + T )$$
83 $$1 + T^{2}$$
89 $$( 1 + T )^{2}$$
97 $$( 1 + T )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$