# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.664 - 0.746i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)2-s + (2.99 + 0.00829i)3-s + 2i·4-s + (3.67 + 3.39i)5-s + (2.99 + 3.00i)6-s + (1.45 − 6.84i)7-s + (−2 + 2i)8-s + (8.99 + 0.0497i)9-s + (0.274 + 7.06i)10-s − 6.08i·11-s + (−0.0165 + 5.99i)12-s + (−4.00 + 4.00i)13-s + (8.30 − 5.38i)14-s + (10.9 + 10.2i)15-s − 4·16-s + (−14.8 + 14.8i)17-s + ⋯
 L(s)  = 1 + (0.5 + 0.5i)2-s + (0.999 + 0.00276i)3-s + 0.5i·4-s + (0.734 + 0.679i)5-s + (0.498 + 0.501i)6-s + (0.208 − 0.978i)7-s + (−0.250 + 0.250i)8-s + (0.999 + 0.00553i)9-s + (0.0274 + 0.706i)10-s − 0.553i·11-s + (−0.00138 + 0.499i)12-s + (−0.307 + 0.307i)13-s + (0.593 − 0.384i)14-s + (0.732 + 0.681i)15-s − 0.250·16-s + (−0.871 + 0.871i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.664 - 0.746i$ motivic weight = $$2$$ character : $\chi_{210} (83, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.664 - 0.746i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.69117 + 1.20733i$$ $$L(\frac12)$$ $$\approx$$ $$2.69117 + 1.20733i$$ $$L(2)$$ 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 + (-2.99 - 0.00829i)T$$
5 $$1 + (-3.67 - 3.39i)T$$
7 $$1 + (-1.45 + 6.84i)T$$
good11 $$1 + 6.08iT - 121T^{2}$$
13 $$1 + (4.00 - 4.00i)T - 169iT^{2}$$
17 $$1 + (14.8 - 14.8i)T - 289iT^{2}$$
19 $$1 + 20.4T + 361T^{2}$$
23 $$1 + (-20.6 + 20.6i)T - 529iT^{2}$$
29 $$1 + 19.5T + 841T^{2}$$
31 $$1 + 4.36iT - 961T^{2}$$
37 $$1 + (1.64 - 1.64i)T - 1.36e3iT^{2}$$
41 $$1 - 42.2T + 1.68e3T^{2}$$
43 $$1 + (45.0 + 45.0i)T + 1.84e3iT^{2}$$
47 $$1 + (36.6 - 36.6i)T - 2.20e3iT^{2}$$
53 $$1 + (0.652 - 0.652i)T - 2.80e3iT^{2}$$
59 $$1 + 4.02iT - 3.48e3T^{2}$$
61 $$1 + 65.2iT - 3.72e3T^{2}$$
67 $$1 + (-59.7 + 59.7i)T - 4.48e3iT^{2}$$
71 $$1 - 122. iT - 5.04e3T^{2}$$
73 $$1 + (-13.1 + 13.1i)T - 5.32e3iT^{2}$$
79 $$1 + 126. iT - 6.24e3T^{2}$$
83 $$1 + (12.2 + 12.2i)T + 6.88e3iT^{2}$$
89 $$1 + 97.2iT - 7.92e3T^{2}$$
97 $$1 + (60.6 + 60.6i)T + 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}