# Properties

 Degree 2 Conductor $2^{7}$ Sign $0.0172 + 0.999i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (11.5 − 11.5i)3-s + (14.6 − 14.6i)5-s + 24.0·7-s − 184. i·9-s + (61.7 + 61.7i)11-s + (37.5 + 37.5i)13-s − 336. i·15-s + 96.8·17-s + (−156. + 156. i)19-s + (276. − 276. i)21-s − 959.·23-s + 198. i·25-s + (−1.19e3 − 1.19e3i)27-s + (350. + 350. i)29-s − 237. i·31-s + ⋯
 L(s)  = 1 + (1.28 − 1.28i)3-s + (0.584 − 0.584i)5-s + 0.490·7-s − 2.27i·9-s + (0.510 + 0.510i)11-s + (0.222 + 0.222i)13-s − 1.49i·15-s + 0.335·17-s + (−0.434 + 0.434i)19-s + (0.627 − 0.627i)21-s − 1.81·23-s + 0.317i·25-s + (−1.63 − 1.63i)27-s + (0.416 + 0.416i)29-s − 0.247i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0172 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0172 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$128$$    =    $$2^{7}$$ $$\varepsilon$$ = $0.0172 + 0.999i$ motivic weight = $$4$$ character : $\chi_{128} (95, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 128,\ (\ :2),\ 0.0172 + 0.999i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$2.21371 - 2.17582i$$ $$L(\frac12)$$ $$\approx$$ $$2.21371 - 2.17582i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + (-11.5 + 11.5i)T - 81iT^{2}$$
5 $$1 + (-14.6 + 14.6i)T - 625iT^{2}$$
7 $$1 - 24.0T + 2.40e3T^{2}$$
11 $$1 + (-61.7 - 61.7i)T + 1.46e4iT^{2}$$
13 $$1 + (-37.5 - 37.5i)T + 2.85e4iT^{2}$$
17 $$1 - 96.8T + 8.35e4T^{2}$$
19 $$1 + (156. - 156. i)T - 1.30e5iT^{2}$$
23 $$1 + 959.T + 2.79e5T^{2}$$
29 $$1 + (-350. - 350. i)T + 7.07e5iT^{2}$$
31 $$1 + 237. iT - 9.23e5T^{2}$$
37 $$1 + (-560. + 560. i)T - 1.87e6iT^{2}$$
41 $$1 + 1.80e3iT - 2.82e6T^{2}$$
43 $$1 + (-206. - 206. i)T + 3.41e6iT^{2}$$
47 $$1 - 1.59e3iT - 4.87e6T^{2}$$
53 $$1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2}$$
59 $$1 + (-2.35e3 - 2.35e3i)T + 1.21e7iT^{2}$$
61 $$1 + (-4.44e3 - 4.44e3i)T + 1.38e7iT^{2}$$
67 $$1 + (3.99e3 - 3.99e3i)T - 2.01e7iT^{2}$$
71 $$1 + 4.92e3T + 2.54e7T^{2}$$
73 $$1 + 2.65e3iT - 2.83e7T^{2}$$
79 $$1 - 8.79e3iT - 3.89e7T^{2}$$
83 $$1 + (228. - 228. i)T - 4.74e7iT^{2}$$
89 $$1 - 1.05e4iT - 6.27e7T^{2}$$
97 $$1 - 1.10e4T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}