# Properties

 Degree $2$ Conductor $128$ Sign $-0.405 + 0.914i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.31 − 3.18i)3-s + (−0.659 − 1.59i)5-s + (−9.54 − 9.54i)7-s + (−2.03 − 2.03i)9-s + (3.96 + 9.57i)11-s + (1.91 − 4.63i)13-s − 5.93·15-s − 15.3i·17-s + (−0.827 − 0.342i)19-s + (−42.9 + 17.8i)21-s + (12.9 − 12.9i)23-s + (15.5 − 15.5i)25-s + (19.5 − 8.07i)27-s + (23.7 + 9.85i)29-s + 25.1i·31-s + ⋯
 L(s)  = 1 + (0.439 − 1.06i)3-s + (−0.131 − 0.318i)5-s + (−1.36 − 1.36i)7-s + (−0.225 − 0.225i)9-s + (0.360 + 0.870i)11-s + (0.147 − 0.356i)13-s − 0.395·15-s − 0.900i·17-s + (−0.0435 − 0.0180i)19-s + (−2.04 + 0.847i)21-s + (0.561 − 0.561i)23-s + (0.623 − 0.623i)25-s + (0.722 − 0.299i)27-s + (0.820 + 0.339i)29-s + 0.811i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-0.405 + 0.914i$ Motivic weight: $$2$$ Character: $\chi_{128} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :1),\ -0.405 + 0.914i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.725339 - 1.11511i$$ $$L(\frac12)$$ $$\approx$$ $$0.725339 - 1.11511i$$ $$L(2)$$ 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 + (-1.31 + 3.18i)T + (-6.36 - 6.36i)T^{2}$$
5 $$1 + (0.659 + 1.59i)T + (-17.6 + 17.6i)T^{2}$$
7 $$1 + (9.54 + 9.54i)T + 49iT^{2}$$
11 $$1 + (-3.96 - 9.57i)T + (-85.5 + 85.5i)T^{2}$$
13 $$1 + (-1.91 + 4.63i)T + (-119. - 119. i)T^{2}$$
17 $$1 + 15.3iT - 289T^{2}$$
19 $$1 + (0.827 + 0.342i)T + (255. + 255. i)T^{2}$$
23 $$1 + (-12.9 + 12.9i)T - 529iT^{2}$$
29 $$1 + (-23.7 - 9.85i)T + (594. + 594. i)T^{2}$$
31 $$1 - 25.1iT - 961T^{2}$$
37 $$1 + (-13.6 - 32.8i)T + (-968. + 968. i)T^{2}$$
41 $$1 + (32.9 + 32.9i)T + 1.68e3iT^{2}$$
43 $$1 + (-17.9 - 43.3i)T + (-1.30e3 + 1.30e3i)T^{2}$$
47 $$1 + 20.1T + 2.20e3T^{2}$$
53 $$1 + (-35.0 + 14.5i)T + (1.98e3 - 1.98e3i)T^{2}$$
59 $$1 + (-60.6 + 25.1i)T + (2.46e3 - 2.46e3i)T^{2}$$
61 $$1 + (27.9 + 11.5i)T + (2.63e3 + 2.63e3i)T^{2}$$
67 $$1 + (1.13 - 2.73i)T + (-3.17e3 - 3.17e3i)T^{2}$$
71 $$1 + (-45.6 - 45.6i)T + 5.04e3iT^{2}$$
73 $$1 + (29.1 + 29.1i)T + 5.32e3iT^{2}$$
79 $$1 + 3.27T + 6.24e3T^{2}$$
83 $$1 + (56.7 + 23.5i)T + (4.87e3 + 4.87e3i)T^{2}$$
89 $$1 + (44.5 - 44.5i)T - 7.92e3iT^{2}$$
97 $$1 + 106.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$