# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.514 + 0.857i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s − 2.44i·6-s + (−2.59 + 6.50i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (2.73 − 1.58i)10-s + (−5.13 − 8.89i)11-s + (2.99 + 1.73i)12-s − 7.02i·13-s + (−6.12 − 7.77i)14-s + 3.87·15-s + (−2.00 + 3.46i)16-s + (27.4 − 15.8i)17-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s − 0.408i·6-s + (−0.370 + 0.928i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.273 − 0.158i)10-s + (−0.466 − 0.808i)11-s + (0.249 + 0.144i)12-s − 0.540i·13-s + (−0.437 − 0.555i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (1.61 − 0.933i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.514 + 0.857i$ motivic weight = $$2$$ character : $\chi_{210} (61, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.514 + 0.857i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.482673 - 0.273143i$$ $$L(\frac12)$$ $$\approx$$ $$0.482673 - 0.273143i$$ $$L(2)$$ 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 + (0.707 - 1.22i)T$$
3 $$1 + (1.5 - 0.866i)T$$
5 $$1 + (1.93 + 1.11i)T$$
7 $$1 + (2.59 - 6.50i)T$$
good11 $$1 + (5.13 + 8.89i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + 7.02iT - 169T^{2}$$
17 $$1 + (-27.4 + 15.8i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (-11.8 + 20.5i)T + (-264.5 - 458. i)T^{2}$$
29 $$1 - 9.19T + 841T^{2}$$
31 $$1 + (-17.4 + 10.0i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + (24.0 - 41.7i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 + 65.1iT - 1.68e3T^{2}$$
43 $$1 + 3.03T + 1.84e3T^{2}$$
47 $$1 + (53.6 + 30.9i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (0.690 + 1.19i)T + (-1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (95.1 - 54.9i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (34.3 + 19.8i)T + (1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (7.95 + 13.7i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 53.3T + 5.04e3T^{2}$$
73 $$1 + (-62.6 + 36.1i)T + (2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (53.2 - 92.1i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 - 49.4iT - 6.88e3T^{2}$$
89 $$1 + (142. + 82.4i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 - 49.4iT - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}