# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.993 + 0.116i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.99 + 0.125i)2-s + (3.96 + 0.5i)4-s + (−4.94 − 4.94i)7-s + (7.85 + 1.49i)8-s + (6.36 − 6.36i)9-s + (20.1 + 8.36i)11-s + (−9.26 − 10.5i)14-s + (15.5 + 3.96i)16-s + (13.5 − 11.9i)18-s + (39.2 + 19.2i)22-s + (−30.1 + 30.1i)23-s + (−17.6 − 17.6i)25-s + (−17.1 − 22.1i)28-s + (−37.1 + 15.3i)29-s + (30.4 + 9.86i)32-s + ⋯
 L(s)  = 1 + (0.998 + 0.0626i)2-s + (0.992 + 0.125i)4-s + (−0.707 − 0.707i)7-s + (0.982 + 0.186i)8-s + (0.707 − 0.707i)9-s + (1.83 + 0.760i)11-s + (−0.661 − 0.750i)14-s + (0.968 + 0.248i)16-s + (0.750 − 0.661i)18-s + (1.78 + 0.873i)22-s + (−1.31 + 1.31i)23-s + (−0.707 − 0.707i)25-s + (−0.613 − 0.789i)28-s + (−1.28 + 0.530i)29-s + (0.951 + 0.308i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.993 + 0.116i$ motivic weight = $$2$$ character : $\chi_{224} (181, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.993 + 0.116i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.95935 - 0.173275i$$ $$L(\frac12)$$ $$\approx$$ $$2.95935 - 0.173275i$$ $$L(2)$$ 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 + (-1.99 - 0.125i)T$$
7 $$1 + (4.94 + 4.94i)T$$
good3 $$1 + (-6.36 + 6.36i)T^{2}$$
5 $$1 + (17.6 + 17.6i)T^{2}$$
11 $$1 + (-20.1 - 8.36i)T + (85.5 + 85.5i)T^{2}$$
13 $$1 + (119. - 119. i)T^{2}$$
17 $$1 + 289T^{2}$$
19 $$1 + (255. - 255. i)T^{2}$$
23 $$1 + (30.1 - 30.1i)T - 529iT^{2}$$
29 $$1 + (37.1 - 15.3i)T + (594. - 594. i)T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 + (-22.7 + 54.8i)T + (-968. - 968. i)T^{2}$$
41 $$1 + 1.68e3iT^{2}$$
43 $$1 + (27.0 + 11.2i)T + (1.30e3 + 1.30e3i)T^{2}$$
47 $$1 + 2.20e3T^{2}$$
53 $$1 + (32.2 + 13.3i)T + (1.98e3 + 1.98e3i)T^{2}$$
59 $$1 + (2.46e3 + 2.46e3i)T^{2}$$
61 $$1 + (-2.63e3 + 2.63e3i)T^{2}$$
67 $$1 + (123. - 51.0i)T + (3.17e3 - 3.17e3i)T^{2}$$
71 $$1 + (-59.8 - 59.8i)T + 5.04e3iT^{2}$$
73 $$1 + 5.32e3iT^{2}$$
79 $$1 - 156. iT - 6.24e3T^{2}$$
83 $$1 + (4.87e3 - 4.87e3i)T^{2}$$
89 $$1 - 7.92e3iT^{2}$$
97 $$1 - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}