# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.728 + 0.685i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.22·3-s − 6.26i·5-s − 2.64i·7-s + 18.2·9-s − 9.80·11-s − 2.41i·13-s − 32.7i·15-s + 6.89·17-s − 2.77·19-s − 13.8i·21-s + 42.8i·23-s − 14.2·25-s + 48.5·27-s + 37.3i·29-s − 7.16i·31-s + ⋯
 L(s)  = 1 + 1.74·3-s − 1.25i·5-s − 0.377i·7-s + 2.03·9-s − 0.891·11-s − 0.185i·13-s − 2.18i·15-s + 0.405·17-s − 0.146·19-s − 0.658i·21-s + 1.86i·23-s − 0.571·25-s + 1.79·27-s + 1.28i·29-s − 0.231i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.728 + 0.685i$ motivic weight = $$2$$ character : $\chi_{224} (15, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.728 + 0.685i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.43180 - 0.964640i$$ $$L(\frac12)$$ $$\approx$$ $$2.43180 - 0.964640i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + 2.64iT$$
good3 $$1 - 5.22T + 9T^{2}$$
5 $$1 + 6.26iT - 25T^{2}$$
11 $$1 + 9.80T + 121T^{2}$$
13 $$1 + 2.41iT - 169T^{2}$$
17 $$1 - 6.89T + 289T^{2}$$
19 $$1 + 2.77T + 361T^{2}$$
23 $$1 - 42.8iT - 529T^{2}$$
29 $$1 - 37.3iT - 841T^{2}$$
31 $$1 + 7.16iT - 961T^{2}$$
37 $$1 - 0.202iT - 1.36e3T^{2}$$
41 $$1 - 63.5T + 1.68e3T^{2}$$
43 $$1 - 35.3T + 1.84e3T^{2}$$
47 $$1 + 37.9iT - 2.20e3T^{2}$$
53 $$1 - 54.6iT - 2.80e3T^{2}$$
59 $$1 + 104.T + 3.48e3T^{2}$$
61 $$1 + 43.7iT - 3.72e3T^{2}$$
67 $$1 + 31.1T + 4.48e3T^{2}$$
71 $$1 - 23.1iT - 5.04e3T^{2}$$
73 $$1 + 69.2T + 5.32e3T^{2}$$
79 $$1 - 19.9iT - 6.24e3T^{2}$$
83 $$1 - 5.11T + 6.88e3T^{2}$$
89 $$1 + 17.9T + 7.92e3T^{2}$$
97 $$1 - 12.4T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}