# Properties

 Degree 2 Conductor $2^{7}$ Sign $-0.851 - 0.524i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.63 − 4.63i)3-s + (29.2 + 29.2i)5-s − 59.6·7-s − 38.0i·9-s + (−18.0 + 18.0i)11-s + (−50.7 + 50.7i)13-s − 270. i·15-s − 223.·17-s + (14.7 + 14.7i)19-s + (276. + 276. i)21-s − 739.·23-s + 1.08e3i·25-s + (−551. + 551. i)27-s + (−938. + 938. i)29-s + 938. i·31-s + ⋯
 L(s)  = 1 + (−0.515 − 0.515i)3-s + (1.16 + 1.16i)5-s − 1.21·7-s − 0.469i·9-s + (−0.149 + 0.149i)11-s + (−0.300 + 0.300i)13-s − 1.20i·15-s − 0.774·17-s + (0.0408 + 0.0408i)19-s + (0.626 + 0.626i)21-s − 1.39·23-s + 1.72i·25-s + (−0.756 + 0.756i)27-s + (−1.11 + 1.11i)29-s + 0.976i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$128$$    =    $$2^{7}$$ $$\varepsilon$$ = $-0.851 - 0.524i$ motivic weight = $$4$$ character : $\chi_{128} (31, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 128,\ (\ :2),\ -0.851 - 0.524i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.117794 + 0.415962i$$ $$L(\frac12)$$ $$\approx$$ $$0.117794 + 0.415962i$$ $$L(3)$$ 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 + (4.63 + 4.63i)T + 81iT^{2}$$
5 $$1 + (-29.2 - 29.2i)T + 625iT^{2}$$
7 $$1 + 59.6T + 2.40e3T^{2}$$
11 $$1 + (18.0 - 18.0i)T - 1.46e4iT^{2}$$
13 $$1 + (50.7 - 50.7i)T - 2.85e4iT^{2}$$
17 $$1 + 223.T + 8.35e4T^{2}$$
19 $$1 + (-14.7 - 14.7i)T + 1.30e5iT^{2}$$
23 $$1 + 739.T + 2.79e5T^{2}$$
29 $$1 + (938. - 938. i)T - 7.07e5iT^{2}$$
31 $$1 - 938. iT - 9.23e5T^{2}$$
37 $$1 + (263. + 263. i)T + 1.87e6iT^{2}$$
41 $$1 - 248. iT - 2.82e6T^{2}$$
43 $$1 + (-1.03e3 + 1.03e3i)T - 3.41e6iT^{2}$$
47 $$1 + 2.01e3iT - 4.87e6T^{2}$$
53 $$1 + (833. + 833. i)T + 7.89e6iT^{2}$$
59 $$1 + (2.22e3 - 2.22e3i)T - 1.21e7iT^{2}$$
61 $$1 + (-341. + 341. i)T - 1.38e7iT^{2}$$
67 $$1 + (-4.84e3 - 4.84e3i)T + 2.01e7iT^{2}$$
71 $$1 - 4.18e3T + 2.54e7T^{2}$$
73 $$1 + 9.07e3iT - 2.83e7T^{2}$$
79 $$1 + 735. iT - 3.89e7T^{2}$$
83 $$1 + (-1.44e3 - 1.44e3i)T + 4.74e7iT^{2}$$
89 $$1 + 5.07e3iT - 6.27e7T^{2}$$
97 $$1 + 2.52e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}