Properties

 Degree $2$ Conductor $128$ Sign $-i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 2.82i·3-s − 5.00·9-s + 2.82i·11-s + 6·17-s − 8.48i·19-s + 5·25-s − 5.65i·27-s − 8.00·33-s − 6·41-s − 8.48i·43-s − 7·49-s + 16.9i·51-s + 24·57-s + 14.1i·59-s − 8.48i·67-s + ⋯
 L(s)  = 1 + 1.63i·3-s − 1.66·9-s + 0.852i·11-s + 1.45·17-s − 1.94i·19-s + 25-s − 1.08i·27-s − 1.39·33-s − 0.937·41-s − 1.29i·43-s − 49-s + 2.37i·51-s + 3.17·57-s + 1.84i·59-s − 1.03i·67-s + ⋯

Functional equation

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

Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-i$ Motivic weight: $$1$$ Character: $\chi_{128} (65, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :1/2),\ -i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.748881 + 0.748881i$$ $$L(\frac12)$$ $$\approx$$ $$0.748881 + 0.748881i$$ $$L(\frac{3}{2})$$ 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 - 2.82iT - 3T^{2}$$
5 $$1 - 5T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 - 2.82iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 6T + 17T^{2}$$
19 $$1 + 8.48iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 8.48iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 - 14.1iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 8.48iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 2T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 2.82iT - 83T^{2}$$
89 $$1 + 18T + 89T^{2}$$
97 $$1 + 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$