# Properties

 Degree $2$ Conductor $128$ Sign $-0.977 - 0.209i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.21 − 7.75i)3-s + (−13.6 + 5.65i)5-s + (−9.07 + 9.07i)7-s + (−30.7 − 30.7i)9-s + (−16.7 − 40.3i)11-s + (−38.4 − 15.9i)13-s + 123. i·15-s + 92.5i·17-s + (16.6 + 6.90i)19-s + (41.2 + 99.5i)21-s + (−95.6 − 95.6i)23-s + (65.9 − 65.9i)25-s + (−127. + 52.8i)27-s + (19.3 − 46.7i)29-s + 38.1·31-s + ⋯
 L(s)  = 1 + (0.618 − 1.49i)3-s + (−1.22 + 0.505i)5-s + (−0.490 + 0.490i)7-s + (−1.13 − 1.13i)9-s + (−0.458 − 1.10i)11-s + (−0.819 − 0.339i)13-s + 2.13i·15-s + 1.31i·17-s + (0.201 + 0.0834i)19-s + (0.428 + 1.03i)21-s + (−0.866 − 0.866i)23-s + (0.527 − 0.527i)25-s + (−0.909 + 0.376i)27-s + (0.123 − 0.299i)29-s + 0.221·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-0.977 - 0.209i$ Motivic weight: $$3$$ Character: $\chi_{128} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :3/2),\ -0.977 - 0.209i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.0598338 + 0.566231i$$ $$L(\frac12)$$ $$\approx$$ $$0.0598338 + 0.566231i$$ $$L(\frac{5}{2})$$ 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 + (-3.21 + 7.75i)T + (-19.0 - 19.0i)T^{2}$$
5 $$1 + (13.6 - 5.65i)T + (88.3 - 88.3i)T^{2}$$
7 $$1 + (9.07 - 9.07i)T - 343iT^{2}$$
11 $$1 + (16.7 + 40.3i)T + (-941. + 941. i)T^{2}$$
13 $$1 + (38.4 + 15.9i)T + (1.55e3 + 1.55e3i)T^{2}$$
17 $$1 - 92.5iT - 4.91e3T^{2}$$
19 $$1 + (-16.6 - 6.90i)T + (4.85e3 + 4.85e3i)T^{2}$$
23 $$1 + (95.6 + 95.6i)T + 1.21e4iT^{2}$$
29 $$1 + (-19.3 + 46.7i)T + (-1.72e4 - 1.72e4i)T^{2}$$
31 $$1 - 38.1T + 2.97e4T^{2}$$
37 $$1 + (-227. + 94.2i)T + (3.58e4 - 3.58e4i)T^{2}$$
41 $$1 + (279. + 279. i)T + 6.89e4iT^{2}$$
43 $$1 + (112. + 270. i)T + (-5.62e4 + 5.62e4i)T^{2}$$
47 $$1 + 321. iT - 1.03e5T^{2}$$
53 $$1 + (52.5 + 126. i)T + (-1.05e5 + 1.05e5i)T^{2}$$
59 $$1 + (-332. + 137. i)T + (1.45e5 - 1.45e5i)T^{2}$$
61 $$1 + (-33.6 + 81.3i)T + (-1.60e5 - 1.60e5i)T^{2}$$
67 $$1 + (108. - 262. i)T + (-2.12e5 - 2.12e5i)T^{2}$$
71 $$1 + (272. - 272. i)T - 3.57e5iT^{2}$$
73 $$1 + (-372. - 372. i)T + 3.89e5iT^{2}$$
79 $$1 + 244. iT - 4.93e5T^{2}$$
83 $$1 + (1.23e3 + 509. i)T + (4.04e5 + 4.04e5i)T^{2}$$
89 $$1 + (-216. + 216. i)T - 7.04e5iT^{2}$$
97 $$1 - 779.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$