# Properties

 Degree $2$ Conductor $128$ Sign $0.159 + 0.987i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.0461 + 0.0461i)3-s + (8.04 + 8.04i)5-s − 49.8·7-s − 80.9i·9-s + (84.2 − 84.2i)11-s + (−19.4 + 19.4i)13-s + 0.743i·15-s + 437.·17-s + (−349. − 349. i)19-s + (−2.30 − 2.30i)21-s − 404.·23-s − 495. i·25-s + (7.48 − 7.48i)27-s + (1.03e3 − 1.03e3i)29-s − 1.50e3i·31-s + ⋯
 L(s)  = 1 + (0.00513 + 0.00513i)3-s + (0.321 + 0.321i)5-s − 1.01·7-s − 0.999i·9-s + (0.696 − 0.696i)11-s + (−0.115 + 0.115i)13-s + 0.00330i·15-s + 1.51·17-s + (−0.966 − 0.966i)19-s + (−0.00522 − 0.00522i)21-s − 0.765·23-s − 0.792i·25-s + (0.0102 − 0.0102i)27-s + (1.22 − 1.22i)29-s − 1.56i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $0.159 + 0.987i$ Motivic weight: $$4$$ Character: $\chi_{128} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :2),\ 0.159 + 0.987i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.07797 - 0.917720i$$ $$L(\frac12)$$ $$\approx$$ $$1.07797 - 0.917720i$$ $$L(3)$$ 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 + (-0.0461 - 0.0461i)T + 81iT^{2}$$
5 $$1 + (-8.04 - 8.04i)T + 625iT^{2}$$
7 $$1 + 49.8T + 2.40e3T^{2}$$
11 $$1 + (-84.2 + 84.2i)T - 1.46e4iT^{2}$$
13 $$1 + (19.4 - 19.4i)T - 2.85e4iT^{2}$$
17 $$1 - 437.T + 8.35e4T^{2}$$
19 $$1 + (349. + 349. i)T + 1.30e5iT^{2}$$
23 $$1 + 404.T + 2.79e5T^{2}$$
29 $$1 + (-1.03e3 + 1.03e3i)T - 7.07e5iT^{2}$$
31 $$1 + 1.50e3iT - 9.23e5T^{2}$$
37 $$1 + (-434. - 434. i)T + 1.87e6iT^{2}$$
41 $$1 - 696. iT - 2.82e6T^{2}$$
43 $$1 + (917. - 917. i)T - 3.41e6iT^{2}$$
47 $$1 - 111. iT - 4.87e6T^{2}$$
53 $$1 + (1.04e3 + 1.04e3i)T + 7.89e6iT^{2}$$
59 $$1 + (-1.71e3 + 1.71e3i)T - 1.21e7iT^{2}$$
61 $$1 + (3.71e3 - 3.71e3i)T - 1.38e7iT^{2}$$
67 $$1 + (-1.85e3 - 1.85e3i)T + 2.01e7iT^{2}$$
71 $$1 + 1.16e3T + 2.54e7T^{2}$$
73 $$1 + 905. iT - 2.83e7T^{2}$$
79 $$1 - 5.86e3iT - 3.89e7T^{2}$$
83 $$1 + (-7.56e3 - 7.56e3i)T + 4.74e7iT^{2}$$
89 $$1 - 6.43e3iT - 6.27e7T^{2}$$
97 $$1 + 413.T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$