# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 5$ Sign $0.850 - 0.525i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s + (−1 + i)13-s + (1 + i)17-s + 25-s + 2i·29-s + (−1 − i)37-s + 2·41-s − i·49-s + (1 − i)53-s + (−1 + i)65-s + (−1 + i)73-s + (1 + i)85-s − 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯
 L(s)  = 1 + 5-s + (−1 + i)13-s + (1 + i)17-s + 25-s + 2i·29-s + (−1 − i)37-s + 2·41-s − i·49-s + (1 − i)53-s + (−1 + i)65-s + (−1 + i)73-s + (1 + i)85-s − 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2880$$    =    $$2^{6} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $0.850 - 0.525i$ motivic weight = $$0$$ character : $\chi_{2880} (577, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2880,\ (\ :0),\ 0.850 - 0.525i)$ $L(\frac{1}{2})$ $\approx$ $1.453638547$ $L(\frac12)$ $\approx$ $1.453638547$ $L(1)$ 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 - T$$
good7 $$1 + iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
17 $$1 + (-1 - i)T + iT^{2}$$
19 $$1 - T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 - 2iT - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (1 + i)T + iT^{2}$$
41 $$1 - 2T + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + (-1 + i)T - iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (1 - i)T - iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + 2iT - T^{2}$$
97 $$1 + (1 + i)T + iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}