# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5^{2} \cdot 7$ Sign $0.669 - 0.742i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (−1.73 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.5 + 4.33i)11-s + (0.866 + 0.499i)12-s + 5i·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s + (−0.866 + 0.499i)18-s + (−3.5 + 6.06i)19-s + ⋯
 L(s)  = 1 + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (−0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.753 + 1.30i)11-s + (0.249 + 0.144i)12-s + 1.38i·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s + (−0.204 + 0.117i)18-s + (−0.802 + 1.39i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.669 - 0.742i$ motivic weight = $$1$$ character : $\chi_{1050} (499, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1050,\ (\ :1/2),\ 0.669 - 0.742i)$$ $$L(1)$$ $$\approx$$ $$1.070040171$$ $$L(\frac12)$$ $$\approx$$ $$1.070040171$$ $$L(\frac{3}{2})$$ 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 + (-0.866 + 0.5i)T$$
5 $$1$$
7 $$1 + (1.73 + 2i)T$$
good11 $$1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 - 5iT - 13T^{2}$$
17 $$1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 - 5T + 41T^{2}$$
43 $$1 + 12iT - 43T^{2}$$
47 $$1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 - 2T + 71T^{2}$$
73 $$1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{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}