# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 7$ Sign $-0.992 + 0.120i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 − 0.5i)2-s + (0.0350 − 1.73i)3-s + (0.499 − 0.866i)4-s − 5-s + (−0.835 − 1.51i)6-s + (−1.17 − 2.36i)7-s − 0.999i·8-s + (−2.99 − 0.121i)9-s + (−0.866 + 0.5i)10-s − 0.751i·11-s + (−1.48 − 0.896i)12-s + (−2.72 + 1.57i)13-s + (−2.20 − 1.46i)14-s + (−0.0350 + 1.73i)15-s + (−0.5 − 0.866i)16-s + (0.433 + 0.750i)17-s + ⋯
 L(s)  = 1 + (0.612 − 0.353i)2-s + (0.0202 − 0.999i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.341 − 0.619i)6-s + (−0.445 − 0.895i)7-s − 0.353i·8-s + (−0.999 − 0.0405i)9-s + (−0.273 + 0.158i)10-s − 0.226i·11-s + (−0.427 − 0.258i)12-s + (−0.754 + 0.435i)13-s + (−0.589 − 0.390i)14-s + (−0.00906 + 0.447i)15-s + (−0.125 − 0.216i)16-s + (0.105 + 0.181i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$630$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.992 + 0.120i$ motivic weight = $$1$$ character : $\chi_{630} (551, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 630,\ (\ :1/2),\ -0.992 + 0.120i)$$ $$L(1)$$ $$\approx$$ $$0.0829468 - 1.37549i$$ $$L(\frac12)$$ $$\approx$$ $$0.0829468 - 1.37549i$$ $$L(\frac{3}{2})$$ 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 + (-0.866 + 0.5i)T$$
3 $$1 + (-0.0350 + 1.73i)T$$
5 $$1 + T$$
7 $$1 + (1.17 + 2.36i)T$$
good11 $$1 + 0.751iT - 11T^{2}$$
13 $$1 + (2.72 - 1.57i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + (-0.433 - 0.750i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-1.23 - 0.713i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 5.06iT - 23T^{2}$$
29 $$1 + (5.23 + 3.02i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-5.38 - 3.10i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (-3.60 + 6.24i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-3.69 + 6.40i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (1.72 + 2.99i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (0.790 - 0.456i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (4.21 - 7.30i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-9.30 + 5.37i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 5.43iT - 71T^{2}$$
73 $$1 + (-0.539 + 0.311i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (0.00292 + 0.00506i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-1.01 + 1.76i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (1.22 - 2.12i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-9.23 - 5.33i)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}