# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.998 - 0.0550i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.33 + 2.5i)3-s + (−4.5 − 7.79i)5-s + (6.92 + i)7-s + (8.00 + 13.8i)9-s + (2.59 + 1.5i)11-s + 16·13-s − 45.0i·15-s + (3.5 − 6.06i)17-s + (9.52 − 5.5i)19-s + (27.5 + 21.6i)21-s + (−16.4 + 9.5i)23-s + (−28 + 48.4i)25-s + 35.0i·27-s − 32·29-s + (9.52 + 5.5i)31-s + ⋯
 L(s)  = 1 + (1.44 + 0.833i)3-s + (−0.900 − 1.55i)5-s + (0.989 + 0.142i)7-s + (0.888 + 1.53i)9-s + (0.236 + 0.136i)11-s + 1.23·13-s − 3.00i·15-s + (0.205 − 0.356i)17-s + (0.501 − 0.289i)19-s + (1.30 + 1.03i)21-s + (−0.715 + 0.413i)23-s + (−1.12 + 1.93i)25-s + 1.29i·27-s − 1.10·29-s + (0.307 + 0.177i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0550i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0550i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.998 - 0.0550i$ motivic weight = $$2$$ character : $\chi_{224} (95, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.998 - 0.0550i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.40982 + 0.0663793i$$ $$L(\frac12)$$ $$\approx$$ $$2.40982 + 0.0663793i$$ $$L(2)$$ 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 + (-6.92 - i)T$$
good3 $$1 + (-4.33 - 2.5i)T + (4.5 + 7.79i)T^{2}$$
5 $$1 + (4.5 + 7.79i)T + (-12.5 + 21.6i)T^{2}$$
11 $$1 + (-2.59 - 1.5i)T + (60.5 + 104. i)T^{2}$$
13 $$1 - 16T + 169T^{2}$$
17 $$1 + (-3.5 + 6.06i)T + (-144.5 - 250. i)T^{2}$$
19 $$1 + (-9.52 + 5.5i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (16.4 - 9.5i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + 32T + 841T^{2}$$
31 $$1 + (-9.52 - 5.5i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + 40T + 1.68e3T^{2}$$
43 $$1 + 40iT - 1.84e3T^{2}$$
47 $$1 + (73.6 - 42.5i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (3.5 - 6.06i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (-45.8 - 26.5i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (39.5 + 68.4i)T + (-1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (9.52 + 5.5i)T + (2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + 48iT - 5.04e3T^{2}$$
73 $$1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (30.3 - 17.5i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 8iT - 6.88e3T^{2}$$
89 $$1 + (-48.5 - 84.0i)T + (-3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + 88T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}