# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 9.79·3-s − 8i·5-s − 78.3i·7-s + 14.9·9-s − 107.·11-s − 216i·13-s − 78.3i·15-s − 162·17-s + 440.·19-s − 767. i·21-s + 705. i·23-s + 561·25-s − 646.·27-s − 1.30e3i·29-s + 627. i·31-s + ⋯
 L(s)  = 1 + 1.08·3-s − 0.320i·5-s − 1.59i·7-s + 0.185·9-s − 0.890·11-s − 1.27i·13-s − 0.348i·15-s − 0.560·17-s + 1.22·19-s − 1.74i·21-s + 1.33i·23-s + 0.897·25-s − 0.887·27-s − 1.55i·29-s + 0.652i·31-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$128$$    =    $$2^{7}$$ $$\varepsilon$$ = $i$ motivic weight = $$4$$ character : $\chi_{128} (63, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 128,\ (\ :2),\ i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.51483 - 1.51483i$$ $$L(\frac12)$$ $$\approx$$ $$1.51483 - 1.51483i$$ $$L(3)$$ 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 - 9.79T + 81T^{2}$$
5 $$1 + 8iT - 625T^{2}$$
7 $$1 + 78.3iT - 2.40e3T^{2}$$
11 $$1 + 107.T + 1.46e4T^{2}$$
13 $$1 + 216iT - 2.85e4T^{2}$$
17 $$1 + 162T + 8.35e4T^{2}$$
19 $$1 - 440.T + 1.30e5T^{2}$$
23 $$1 - 705. iT - 2.79e5T^{2}$$
29 $$1 + 1.30e3iT - 7.07e5T^{2}$$
31 $$1 - 627. iT - 9.23e5T^{2}$$
37 $$1 + 1.51e3iT - 1.87e6T^{2}$$
41 $$1 - 1.89e3T + 2.82e6T^{2}$$
43 $$1 - 2.90e3T + 3.41e6T^{2}$$
47 $$1 + 1.41e3iT - 4.87e6T^{2}$$
53 $$1 - 1.97e3iT - 7.89e6T^{2}$$
59 $$1 + 2.26e3T + 1.21e7T^{2}$$
61 $$1 - 2.37e3iT - 1.38e7T^{2}$$
67 $$1 + 1.67e3T + 2.01e7T^{2}$$
71 $$1 - 7.75e3iT - 2.54e7T^{2}$$
73 $$1 - 2.75e3T + 2.83e7T^{2}$$
79 $$1 - 7.99e3iT - 3.89e7T^{2}$$
83 $$1 - 9.33e3T + 4.74e7T^{2}$$
89 $$1 - 2.43e3T + 6.27e7T^{2}$$
97 $$1 - 7.45e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}