# Properties

 Degree 2 Conductor $2^{7}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·3-s + 2·5-s + 4·7-s + 9-s + 2·11-s + 2·13-s − 4·15-s − 2·17-s − 2·19-s − 8·21-s − 4·23-s − 25-s + 4·27-s − 6·29-s − 4·33-s + 8·35-s + 10·37-s − 4·39-s − 6·41-s − 6·43-s + 2·45-s + 8·47-s + 9·49-s + 4·51-s − 6·53-s + 4·55-s + 4·57-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s − 1.74·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.696·33-s + 1.35·35-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$128$$    =    $$2^{7}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{128} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 128,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.9713362299$ $L(\frac12)$ $\approx$ $0.9713362299$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$F_p(T) = 1 - a_p T + p T^2 .$If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 2 T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 6 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 + 6 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 14 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 - 14 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 + 2 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}