# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.806 - 0.591i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.19 − 2.41i)3-s + (−0.0446 + 0.0257i)5-s + (−6.12 + 3.39i)7-s + (7.20 + 12.4i)9-s + (−0.894 + 1.55i)11-s − 5.87i·13-s + 0.249·15-s + (23.0 + 13.2i)17-s + (22.8 − 13.1i)19-s + (33.8 + 0.603i)21-s + (12.8 + 22.2i)23-s + (−12.4 + 21.6i)25-s − 26.1i·27-s − 27.1·29-s + (25.7 + 14.8i)31-s + ⋯
 L(s)  = 1 + (−1.39 − 0.806i)3-s + (−0.00892 + 0.00515i)5-s + (−0.874 + 0.484i)7-s + (0.800 + 1.38i)9-s + (−0.0813 + 0.140i)11-s − 0.452i·13-s + 0.0166·15-s + (1.35 + 0.781i)17-s + (1.20 − 0.693i)19-s + (1.61 + 0.0287i)21-s + (0.558 + 0.966i)23-s + (−0.499 + 0.865i)25-s − 0.969i·27-s − 0.937·29-s + (0.829 + 0.479i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.806 - 0.591i$ motivic weight = $$2$$ character : $\chi_{224} (129, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.806 - 0.591i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.684818 + 0.224106i$$ $$L(\frac12)$$ $$\approx$$ $$0.684818 + 0.224106i$$ $$L(2)$$ 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 + (6.12 - 3.39i)T$$
good3 $$1 + (4.19 + 2.41i)T + (4.5 + 7.79i)T^{2}$$
5 $$1 + (0.0446 - 0.0257i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (0.894 - 1.55i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 + 5.87iT - 169T^{2}$$
17 $$1 + (-23.0 - 13.2i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (-22.8 + 13.1i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (-12.8 - 22.2i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + 27.1T + 841T^{2}$$
31 $$1 + (-25.7 - 14.8i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-30.8 - 53.4i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + 65.7iT - 1.68e3T^{2}$$
43 $$1 + 9.52T + 1.84e3T^{2}$$
47 $$1 + (61.2 - 35.3i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (4.86 - 8.42i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (-54.3 - 31.3i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (66.1 - 38.2i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (51.5 - 89.2i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 90.1T + 5.04e3T^{2}$$
73 $$1 + (28.8 + 16.6i)T + (2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-32.4 - 56.1i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 + 29.1iT - 6.88e3T^{2}$$
89 $$1 + (-18.7 + 10.8i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 - 123. iT - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}