# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.772 - 0.634i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−2.20 − 4.48i)5-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.69 + 2.28i)10-s + 15.1·11-s + (2.44 − 2.44i)12-s + (14.6 + 14.6i)13-s − 3.74i·14-s + (2.79 − 8.19i)15-s − 4·16-s + (15.9 − 15.9i)17-s + ⋯
 L(s)  = 1 + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.440 − 0.897i)5-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.669 + 0.228i)10-s + 1.37·11-s + (0.204 − 0.204i)12-s + (1.12 + 1.12i)13-s − 0.267i·14-s + (0.186 − 0.546i)15-s − 0.250·16-s + (0.940 − 0.940i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.772 - 0.634i$ motivic weight = $$2$$ character : $\chi_{210} (43, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.772 - 0.634i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.29438 + 0.463700i$$ $$L(\frac12)$$ $$\approx$$ $$1.29438 + 0.463700i$$ $$L(2)$$ 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 + (1 - i)T$$
3 $$1 + (-1.22 - 1.22i)T$$
5 $$1 + (2.20 + 4.48i)T$$
7 $$1 + (1.87 - 1.87i)T$$
good11 $$1 - 15.1T + 121T^{2}$$
13 $$1 + (-14.6 - 14.6i)T + 169iT^{2}$$
17 $$1 + (-15.9 + 15.9i)T - 289iT^{2}$$
19 $$1 + 2.05iT - 361T^{2}$$
23 $$1 + (-19.4 - 19.4i)T + 529iT^{2}$$
29 $$1 + 33.1iT - 841T^{2}$$
31 $$1 - 27.8T + 961T^{2}$$
37 $$1 + (30.4 - 30.4i)T - 1.36e3iT^{2}$$
41 $$1 + 26.6T + 1.68e3T^{2}$$
43 $$1 + (-7.89 - 7.89i)T + 1.84e3iT^{2}$$
47 $$1 + (-33.8 + 33.8i)T - 2.20e3iT^{2}$$
53 $$1 + (-5.60 - 5.60i)T + 2.80e3iT^{2}$$
59 $$1 + 5.80iT - 3.48e3T^{2}$$
61 $$1 + 98.2T + 3.72e3T^{2}$$
67 $$1 + (-51.6 + 51.6i)T - 4.48e3iT^{2}$$
71 $$1 + 120.T + 5.04e3T^{2}$$
73 $$1 + (-81.9 - 81.9i)T + 5.32e3iT^{2}$$
79 $$1 + 33.0iT - 6.24e3T^{2}$$
83 $$1 + (97.0 + 97.0i)T + 6.88e3iT^{2}$$
89 $$1 - 34.2iT - 7.92e3T^{2}$$
97 $$1 + (-104. + 104. i)T - 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}