# Properties

 Degree $2$ Conductor $128$ Sign $0.975 - 0.220i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (6.06 − 2.51i)3-s + (−2.91 + 7.02i)5-s + (13.3 + 13.3i)7-s + (11.3 − 11.3i)9-s + (49.6 + 20.5i)11-s + (8.74 + 21.1i)13-s + 49.9i·15-s − 77.7i·17-s + (−53.3 − 128. i)19-s + (114. + 47.5i)21-s + (35.5 − 35.5i)23-s + (47.4 + 47.4i)25-s + (−27.3 + 66.0i)27-s + (−245. + 101. i)29-s + 202.·31-s + ⋯
 L(s)  = 1 + (1.16 − 0.483i)3-s + (−0.260 + 0.628i)5-s + (0.722 + 0.722i)7-s + (0.421 − 0.421i)9-s + (1.36 + 0.563i)11-s + (0.186 + 0.450i)13-s + 0.859i·15-s − 1.10i·17-s + (−0.643 − 1.55i)19-s + (1.19 + 0.494i)21-s + (0.321 − 0.321i)23-s + (0.379 + 0.379i)25-s + (−0.195 + 0.470i)27-s + (−1.57 + 0.650i)29-s + 1.17·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $0.975 - 0.220i$ Motivic weight: $$3$$ Character: $\chi_{128} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :3/2),\ 0.975 - 0.220i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.44767 + 0.273525i$$ $$L(\frac12)$$ $$\approx$$ $$2.44767 + 0.273525i$$ $$L(\frac{5}{2})$$ 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 + (-6.06 + 2.51i)T + (19.0 - 19.0i)T^{2}$$
5 $$1 + (2.91 - 7.02i)T + (-88.3 - 88.3i)T^{2}$$
7 $$1 + (-13.3 - 13.3i)T + 343iT^{2}$$
11 $$1 + (-49.6 - 20.5i)T + (941. + 941. i)T^{2}$$
13 $$1 + (-8.74 - 21.1i)T + (-1.55e3 + 1.55e3i)T^{2}$$
17 $$1 + 77.7iT - 4.91e3T^{2}$$
19 $$1 + (53.3 + 128. i)T + (-4.85e3 + 4.85e3i)T^{2}$$
23 $$1 + (-35.5 + 35.5i)T - 1.21e4iT^{2}$$
29 $$1 + (245. - 101. i)T + (1.72e4 - 1.72e4i)T^{2}$$
31 $$1 - 202.T + 2.97e4T^{2}$$
37 $$1 + (-36.3 + 87.6i)T + (-3.58e4 - 3.58e4i)T^{2}$$
41 $$1 + (36.8 - 36.8i)T - 6.89e4iT^{2}$$
43 $$1 + (185. + 76.8i)T + (5.62e4 + 5.62e4i)T^{2}$$
47 $$1 - 82.9iT - 1.03e5T^{2}$$
53 $$1 + (534. + 221. i)T + (1.05e5 + 1.05e5i)T^{2}$$
59 $$1 + (75.7 - 182. i)T + (-1.45e5 - 1.45e5i)T^{2}$$
61 $$1 + (472. - 195. i)T + (1.60e5 - 1.60e5i)T^{2}$$
67 $$1 + (-102. + 42.5i)T + (2.12e5 - 2.12e5i)T^{2}$$
71 $$1 + (520. + 520. i)T + 3.57e5iT^{2}$$
73 $$1 + (-244. + 244. i)T - 3.89e5iT^{2}$$
79 $$1 + 774. iT - 4.93e5T^{2}$$
83 $$1 + (-23.9 - 57.7i)T + (-4.04e5 + 4.04e5i)T^{2}$$
89 $$1 + (351. + 351. i)T + 7.04e5iT^{2}$$
97 $$1 - 1.30e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$