# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $-0.707 + 0.707i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.85i·3-s − 5.78·5-s − 2.64i·7-s − 25.2·9-s − 3.01i·11-s + 9.78·13-s − 33.8i·15-s − 11.6·17-s + 25.5i·19-s + 15.4·21-s − 26.1i·23-s + 8.42·25-s − 95.4i·27-s + 1.56·29-s + 12.0i·31-s + ⋯
 L(s)  = 1 + 1.95i·3-s − 1.15·5-s − 0.377i·7-s − 2.81·9-s − 0.274i·11-s + 0.752·13-s − 2.25i·15-s − 0.682·17-s + 1.34i·19-s + 0.737·21-s − 1.13i·23-s + 0.337·25-s − 3.53i·27-s + 0.0539·29-s + 0.387i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $-0.707 + 0.707i$ motivic weight = $$2$$ character : $\chi_{224} (127, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ -0.707 + 0.707i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.165724 - 0.400094i$$ $$L(\frac12)$$ $$\approx$$ $$0.165724 - 0.400094i$$ $$L(2)$$ 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 - 5.85iT - 9T^{2}$$
5 $$1 + 5.78T + 25T^{2}$$
11 $$1 + 3.01iT - 121T^{2}$$
13 $$1 - 9.78T + 169T^{2}$$
17 $$1 + 11.6T + 289T^{2}$$
19 $$1 - 25.5iT - 361T^{2}$$
23 $$1 + 26.1iT - 529T^{2}$$
29 $$1 - 1.56T + 841T^{2}$$
31 $$1 - 12.0iT - 961T^{2}$$
37 $$1 + 70.6T + 1.36e3T^{2}$$
41 $$1 + 49.8T + 1.68e3T^{2}$$
43 $$1 - 73.2iT - 1.84e3T^{2}$$
47 $$1 - 44.2iT - 2.20e3T^{2}$$
53 $$1 + 54.2T + 2.80e3T^{2}$$
59 $$1 + 12.4iT - 3.48e3T^{2}$$
61 $$1 - 35.6T + 3.72e3T^{2}$$
67 $$1 - 24.4iT - 4.48e3T^{2}$$
71 $$1 + 11.0iT - 5.04e3T^{2}$$
73 $$1 - 74.3T + 5.32e3T^{2}$$
79 $$1 - 22.8iT - 6.24e3T^{2}$$
83 $$1 - 48.1iT - 6.88e3T^{2}$$
89 $$1 - 67.4T + 7.92e3T^{2}$$
97 $$1 + 7.75T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}