# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $0.978 - 0.205i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)3-s + (4.5 + 2.59i)5-s + (−1 − 6.92i)7-s + (4 − 6.92i)9-s + (8.5 + 14.7i)11-s − 13.8i·13-s + 5.19i·15-s + (12.5 + 21.6i)17-s + (−3.5 + 6.06i)19-s + (5.49 − 4.33i)21-s + (4.5 + 2.59i)23-s + (1 + 1.73i)25-s + 17·27-s + 13.8i·29-s + (28.5 − 16.4i)31-s + ⋯
 L(s)  = 1 + (0.166 + 0.288i)3-s + (0.900 + 0.519i)5-s + (−0.142 − 0.989i)7-s + (0.444 − 0.769i)9-s + (0.772 + 1.33i)11-s − 1.06i·13-s + 0.346i·15-s + (0.735 + 1.27i)17-s + (−0.184 + 0.319i)19-s + (0.261 − 0.206i)21-s + (0.195 + 0.112i)23-s + (0.0400 + 0.0692i)25-s + 0.629·27-s + 0.477i·29-s + (0.919 − 0.530i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $0.978 - 0.205i$ motivic weight = $$2$$ character : $\chi_{224} (79, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 0.978 - 0.205i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.97797 + 0.205177i$$ $$L(\frac12)$$ $$\approx$$ $$1.97797 + 0.205177i$$ $$L(2)$$ 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 + (1 + 6.92i)T$$
good3 $$1 + (-0.5 - 0.866i)T + (-4.5 + 7.79i)T^{2}$$
5 $$1 + (-4.5 - 2.59i)T + (12.5 + 21.6i)T^{2}$$
11 $$1 + (-8.5 - 14.7i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + 13.8iT - 169T^{2}$$
17 $$1 + (-12.5 - 21.6i)T + (-144.5 + 250. i)T^{2}$$
19 $$1 + (3.5 - 6.06i)T + (-180.5 - 312. i)T^{2}$$
23 $$1 + (-4.5 - 2.59i)T + (264.5 + 458. i)T^{2}$$
29 $$1 - 13.8iT - 841T^{2}$$
31 $$1 + (-28.5 + 16.4i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + (7.5 + 4.33i)T + (684.5 + 1.18e3i)T^{2}$$
41 $$1 - 26T + 1.68e3T^{2}$$
43 $$1 + 14T + 1.84e3T^{2}$$
47 $$1 + (43.5 + 25.1i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (79.5 - 45.8i)T + (1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (27.5 + 47.6i)T + (-1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (19.5 + 11.2i)T + (1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (-8.5 - 14.7i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + (59.5 + 103. i)T + (-2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-64.5 - 37.2i)T + (3.12e3 + 5.40e3i)T^{2}$$
83 $$1 + 110T + 6.88e3T^{2}$$
89 $$1 + (35.5 - 61.4i)T + (-3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + 22T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}