# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7^{2}$ Sign $-0.894 + 0.447i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s − i·8-s − 9-s + (1 + 2i)10-s − 5·11-s − i·12-s + i·13-s + (1 + 2i)15-s + 16-s + 2i·17-s − i·18-s − 7·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s − 1.50·11-s − 0.288i·12-s + 0.277i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 1.60·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1470$$    =    $$2 \cdot 3 \cdot 5 \cdot 7^{2}$$ $$\varepsilon$$ = $-0.894 + 0.447i$ motivic weight = $$1$$ character : $\chi_{1470} (589, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1470,\ (\ :1/2),\ -0.894 + 0.447i)$$ $$L(1)$$ $$\approx$$ $$0.6126547694$$ $$L(\frac12)$$ $$\approx$$ $$0.6126547694$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 - iT$$
5 $$1 + (-2 + i)T$$
7 $$1$$
good11 $$1 + 5T + 11T^{2}$$
13 $$1 - iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 7T + 19T^{2}$$
23 $$1 - 3iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 6T + 31T^{2}$$
37 $$1 - 5iT - 37T^{2}$$
41 $$1 + 9T + 41T^{2}$$
43 $$1 - 10iT - 43T^{2}$$
47 $$1 - 13iT - 47T^{2}$$
53 $$1 + iT - 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 + 6iT - 67T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 - 14T + 79T^{2}$$
83 $$1 + 10iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 8iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}