# Properties

 Degree $2$ Conductor $128$ Sign $-0.901 - 0.432i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (9.42 + 9.42i)3-s + (2.84 + 2.84i)5-s − 76.7·7-s + 96.6i·9-s + (−121. + 121. i)11-s + (−27.1 + 27.1i)13-s + 53.6i·15-s − 88.0·17-s + (261. + 261. i)19-s + (−723. − 723. i)21-s + 93.4·23-s − 608. i·25-s + (−147. + 147. i)27-s + (−272. + 272. i)29-s + 1.23e3i·31-s + ⋯
 L(s)  = 1 + (1.04 + 1.04i)3-s + (0.113 + 0.113i)5-s − 1.56·7-s + 1.19i·9-s + (−1.00 + 1.00i)11-s + (−0.160 + 0.160i)13-s + 0.238i·15-s − 0.304·17-s + (0.723 + 0.723i)19-s + (−1.64 − 1.64i)21-s + 0.176·23-s − 0.974i·25-s + (−0.202 + 0.202i)27-s + (−0.324 + 0.324i)29-s + 1.28i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-0.901 - 0.432i$ Motivic weight: $$4$$ Character: $\chi_{128} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :2),\ -0.901 - 0.432i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.318121 + 1.39917i$$ $$L(\frac12)$$ $$\approx$$ $$0.318121 + 1.39917i$$ $$L(3)$$ 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 + (-9.42 - 9.42i)T + 81iT^{2}$$
5 $$1 + (-2.84 - 2.84i)T + 625iT^{2}$$
7 $$1 + 76.7T + 2.40e3T^{2}$$
11 $$1 + (121. - 121. i)T - 1.46e4iT^{2}$$
13 $$1 + (27.1 - 27.1i)T - 2.85e4iT^{2}$$
17 $$1 + 88.0T + 8.35e4T^{2}$$
19 $$1 + (-261. - 261. i)T + 1.30e5iT^{2}$$
23 $$1 - 93.4T + 2.79e5T^{2}$$
29 $$1 + (272. - 272. i)T - 7.07e5iT^{2}$$
31 $$1 - 1.23e3iT - 9.23e5T^{2}$$
37 $$1 + (-1.04e3 - 1.04e3i)T + 1.87e6iT^{2}$$
41 $$1 + 915. iT - 2.82e6T^{2}$$
43 $$1 + (1.11e3 - 1.11e3i)T - 3.41e6iT^{2}$$
47 $$1 - 1.72e3iT - 4.87e6T^{2}$$
53 $$1 + (734. + 734. i)T + 7.89e6iT^{2}$$
59 $$1 + (-1.20e3 + 1.20e3i)T - 1.21e7iT^{2}$$
61 $$1 + (580. - 580. i)T - 1.38e7iT^{2}$$
67 $$1 + (-1.48e3 - 1.48e3i)T + 2.01e7iT^{2}$$
71 $$1 - 5.57e3T + 2.54e7T^{2}$$
73 $$1 + 6.61e3iT - 2.83e7T^{2}$$
79 $$1 + 5.39e3iT - 3.89e7T^{2}$$
83 $$1 + (-2.55e3 - 2.55e3i)T + 4.74e7iT^{2}$$
89 $$1 - 1.09e4iT - 6.27e7T^{2}$$
97 $$1 - 4.71e3T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$