# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 7$ Sign $0.537 + 0.843i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (1.35 − 1.08i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1.35 + 1.08i)6-s + (2.32 + 1.26i)7-s − 8-s + (0.653 − 2.92i)9-s + (0.5 + 0.866i)10-s + (−0.975 + 1.68i)11-s + (1.35 − 1.08i)12-s + (2.18 − 3.77i)13-s + (−2.32 − 1.26i)14-s + (−1.61 − 0.629i)15-s + 16-s + (2.22 + 3.85i)17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.780 − 0.625i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.551 + 0.442i)6-s + (0.878 + 0.478i)7-s − 0.353·8-s + (0.217 − 0.975i)9-s + (0.158 + 0.273i)10-s + (−0.294 + 0.509i)11-s + (0.390 − 0.312i)12-s + (0.605 − 1.04i)13-s + (−0.621 − 0.338i)14-s + (−0.416 − 0.162i)15-s + 0.250·16-s + (0.539 + 0.934i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$630$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.537 + 0.843i$ motivic weight = $$1$$ character : $\chi_{630} (121, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 630,\ (\ :1/2),\ 0.537 + 0.843i)$ $L(1)$ $\approx$ $1.31724 - 0.722948i$ $L(\frac12)$ $\approx$ $1.31724 - 0.722948i$ $L(\frac{3}{2})$ 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 + T$$
3 $$1 + (-1.35 + 1.08i)T$$
5 $$1 + (0.5 + 0.866i)T$$
7 $$1 + (-2.32 - 1.26i)T$$
good11 $$1 + (0.975 - 1.68i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-2.18 + 3.77i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-2.22 - 3.85i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.18 + 3.77i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.957 + 1.65i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-0.725 - 1.25i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 2.78T + 31T^{2}$$
37 $$1 + (1.44 - 2.50i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-4.07 + 7.05i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (1.39 + 2.42i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 6.79T + 47T^{2}$$
53 $$1 + (3.07 + 5.33i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 8.10T + 59T^{2}$$
61 $$1 - 0.523T + 61T^{2}$$
67 $$1 + 11.0T + 67T^{2}$$
71 $$1 - 16.3T + 71T^{2}$$
73 $$1 + (-7.69 - 13.3i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 - 7.55T + 79T^{2}$$
83 $$1 + (-1.48 - 2.58i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-4.62 + 8.01i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (6.19 + 10.7i)T + (-48.5 + 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}