# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.197 + 0.980i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s − i·7-s + (0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (−0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯
 L(s)  = 1 + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s − i·7-s + (0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (−0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.197 + 0.980i$ motivic weight = $$0$$ character : $\chi_{224} (95, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :0),\ 0.197 + 0.980i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.5239175543$$ $$L(\frac12)$$ $$\approx$$ $$0.5239175543$$ $$L(1)$$ 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 + iT$$
good3 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
5 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
37 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + T^{2}$$
43 $$1 - T^{2}$$
47 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
97 $$1 + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}