# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{2} \cdot 5$ Sign $0.367 - 0.930i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.86 − 1.23i)5-s + 0.746i·7-s + 5.36i·11-s − 2.92·13-s + 2.13i·17-s + 1.73i·19-s + 7.49i·23-s + (1.92 − 4.61i)25-s + 6.74i·29-s − 2.64·31-s + (0.925 + 1.38i)35-s − 1.07·37-s + 11.2·41-s − 7.44·43-s − 1.73i·47-s + ⋯
 L(s)  = 1 + (0.832 − 0.554i)5-s + 0.282i·7-s + 1.61i·11-s − 0.811·13-s + 0.517i·17-s + 0.397i·19-s + 1.56i·23-s + (0.385 − 0.922i)25-s + 1.25i·29-s − 0.475·31-s + (0.156 + 0.234i)35-s − 0.176·37-s + 1.76·41-s − 1.13·43-s − 0.252i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1440$$    =    $$2^{5} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $0.367 - 0.930i$ motivic weight = $$1$$ character : $\chi_{1440} (1009, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1440,\ (\ :1/2),\ 0.367 - 0.930i)$$ $$L(1)$$ $$\approx$$ $$1.653025560$$ $$L(\frac12)$$ $$\approx$$ $$1.653025560$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (-1.86 + 1.23i)T$$
good7 $$1 - 0.746iT - 7T^{2}$$
11 $$1 - 5.36iT - 11T^{2}$$
13 $$1 + 2.92T + 13T^{2}$$
17 $$1 - 2.13iT - 17T^{2}$$
19 $$1 - 1.73iT - 19T^{2}$$
23 $$1 - 7.49iT - 23T^{2}$$
29 $$1 - 6.74iT - 29T^{2}$$
31 $$1 + 2.64T + 31T^{2}$$
37 $$1 + 1.07T + 37T^{2}$$
41 $$1 - 11.2T + 41T^{2}$$
43 $$1 + 7.44T + 43T^{2}$$
47 $$1 + 1.73iT - 47T^{2}$$
53 $$1 + 7.72T + 53T^{2}$$
59 $$1 + 6.85iT - 59T^{2}$$
61 $$1 + 6.45iT - 61T^{2}$$
67 $$1 - 7.44T + 67T^{2}$$
71 $$1 - 13.2T + 71T^{2}$$
73 $$1 + 0.690iT - 73T^{2}$$
79 $$1 - 2.64T + 79T^{2}$$
83 $$1 + 5.85T + 83T^{2}$$
89 $$1 - 7.59T + 89T^{2}$$
97 $$1 - 14.1iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}