# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $-0.143 + 0.989i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.89 + 0.649i)2-s + (−5.08 − 2.10i)3-s + (3.15 − 2.45i)4-s + (−0.976 − 2.35i)5-s + (10.9 + 0.684i)6-s + (4.79 + 5.09i)7-s + (−4.37 + 6.69i)8-s + (15.0 + 15.0i)9-s + (3.37 + 3.82i)10-s + (2.51 + 6.07i)11-s + (−21.2 + 5.84i)12-s + (9.18 − 22.1i)13-s + (−12.3 − 6.52i)14-s + 14.0i·15-s + (3.93 − 15.5i)16-s − 12.9·17-s + ⋯
 L(s)  = 1 + (−0.945 + 0.324i)2-s + (−1.69 − 0.702i)3-s + (0.789 − 0.613i)4-s + (−0.195 − 0.471i)5-s + (1.83 + 0.114i)6-s + (0.685 + 0.727i)7-s + (−0.547 + 0.836i)8-s + (1.67 + 1.67i)9-s + (0.337 + 0.382i)10-s + (0.228 + 0.552i)11-s + (−1.77 + 0.486i)12-s + (0.706 − 1.70i)13-s + (−0.884 − 0.466i)14-s + 0.937i·15-s + (0.246 − 0.969i)16-s − 0.761·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $-0.143 + 0.989i$ motivic weight = $$2$$ character : $\chi_{224} (13, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ -0.143 + 0.989i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.294858 - 0.340867i$$ $$L(\frac12)$$ $$\approx$$ $$0.294858 - 0.340867i$$ $$L(2)$$ 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 + (1.89 - 0.649i)T$$
7 $$1 + (-4.79 - 5.09i)T$$
good3 $$1 + (5.08 + 2.10i)T + (6.36 + 6.36i)T^{2}$$
5 $$1 + (0.976 + 2.35i)T + (-17.6 + 17.6i)T^{2}$$
11 $$1 + (-2.51 - 6.07i)T + (-85.5 + 85.5i)T^{2}$$
13 $$1 + (-9.18 + 22.1i)T + (-119. - 119. i)T^{2}$$
17 $$1 + 12.9T + 289T^{2}$$
19 $$1 + (5.85 - 14.1i)T + (-255. - 255. i)T^{2}$$
23 $$1 + (-3.80 - 3.80i)T + 529iT^{2}$$
29 $$1 + (-10.4 + 25.3i)T + (-594. - 594. i)T^{2}$$
31 $$1 + 26.1iT - 961T^{2}$$
37 $$1 + (22.3 - 9.26i)T + (968. - 968. i)T^{2}$$
41 $$1 + (-1.69 + 1.69i)T - 1.68e3iT^{2}$$
43 $$1 + (23.2 + 56.0i)T + (-1.30e3 + 1.30e3i)T^{2}$$
47 $$1 + 63.8T + 2.20e3T^{2}$$
53 $$1 + (29.7 + 71.8i)T + (-1.98e3 + 1.98e3i)T^{2}$$
59 $$1 + (-25.6 - 61.8i)T + (-2.46e3 + 2.46e3i)T^{2}$$
61 $$1 + (-46.9 - 19.4i)T + (2.63e3 + 2.63e3i)T^{2}$$
67 $$1 + (0.333 - 0.805i)T + (-3.17e3 - 3.17e3i)T^{2}$$
71 $$1 + (-48.5 + 48.5i)T - 5.04e3iT^{2}$$
73 $$1 + (-69.1 + 69.1i)T - 5.32e3iT^{2}$$
79 $$1 + 104. iT - 6.24e3T^{2}$$
83 $$1 + (-22.3 + 54.0i)T + (-4.87e3 - 4.87e3i)T^{2}$$
89 $$1 + (47.9 + 47.9i)T + 7.92e3iT^{2}$$
97 $$1 - 8.86iT - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}