# Properties

 Degree 2 Conductor $2 \cdot 19^{2}$ Sign $-0.910 - 0.412i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (1 + 1.73i)9-s − 6·11-s + 0.999·12-s + (−2.5 − 4.33i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + 2·18-s + (0.5 − 0.866i)21-s + (−3 + 5.19i)22-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s − 0.377·7-s − 0.353·8-s + (0.333 + 0.577i)9-s − 1.80·11-s + 0.288·12-s + (−0.693 − 1.20i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.471·18-s + (0.109 − 0.188i)21-s + (−0.639 + 1.10i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$722$$    =    $$2 \cdot 19^{2}$$ $$\varepsilon$$ = $-0.910 - 0.412i$ motivic weight = $$1$$ character : $\chi_{722} (429, \cdot )$ primitive : yes self-dual : no analytic rank = 1 Selberg data = $(2,\ 722,\ (\ :1/2),\ -0.910 - 0.412i)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;19\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
19 $$1$$
good3 $$1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (-2.5 - 4.33i)T^{2}$$
7 $$1 + T + 7T^{2}$$
11 $$1 + 6T + 11T^{2}$$
13 $$1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}$$
23 $$1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}