# Properties

 Degree $2$ Conductor $722$ Sign $0.999 + 0.00130i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (3.75 − 1.36i)5-s + (−0.766 + 0.642i)6-s + (−1.5 − 2.59i)7-s + (0.5 − 0.866i)8-s + (−0.347 − 1.96i)9-s + (0.694 + 3.93i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (0.766 − 0.642i)13-s + (2.81 − 1.02i)14-s + (3.75 + 1.36i)15-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s + ⋯
 L(s)  = 1 + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (1.68 − 0.611i)5-s + (−0.312 + 0.262i)6-s + (−0.566 − 0.981i)7-s + (0.176 − 0.306i)8-s + (−0.115 − 0.656i)9-s + (0.219 + 1.24i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.249i)12-s + (0.212 − 0.178i)13-s + (0.753 − 0.274i)14-s + (0.970 + 0.353i)15-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$722$$    =    $$2 \cdot 19^{2}$$ Sign: $0.999 + 0.00130i$ Motivic weight: $$1$$ Character: $\chi_{722} (245, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 722,\ (\ :1/2),\ 0.999 + 0.00130i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.90907 - 0.00124606i$$ $$L(\frac12)$$ $$\approx$$ $$1.90907 - 0.00124606i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.173 - 0.984i)T$$
19 $$1$$
good3 $$1 + (-0.766 - 0.642i)T + (0.520 + 2.95i)T^{2}$$
5 $$1 + (-3.75 + 1.36i)T + (3.83 - 3.21i)T^{2}$$
7 $$1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-0.766 + 0.642i)T + (2.25 - 12.8i)T^{2}$$
17 $$1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2}$$
23 $$1 + (-0.939 - 0.342i)T + (17.6 + 14.7i)T^{2}$$
29 $$1 + (-0.868 - 4.92i)T + (-27.2 + 9.91i)T^{2}$$
31 $$1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (-6.12 - 5.14i)T + (7.11 + 40.3i)T^{2}$$
43 $$1 + (3.75 - 1.36i)T + (32.9 - 27.6i)T^{2}$$
47 $$1 + (-1.38 - 7.87i)T + (-44.1 + 16.0i)T^{2}$$
53 $$1 + (0.939 + 0.342i)T + (40.6 + 34.0i)T^{2}$$
59 $$1 + (2.60 - 14.7i)T + (-55.4 - 20.1i)T^{2}$$
61 $$1 + (1.87 + 0.684i)T + (46.7 + 39.2i)T^{2}$$
67 $$1 + (0.520 + 2.95i)T + (-62.9 + 22.9i)T^{2}$$
71 $$1 + (-1.87 + 0.684i)T + (54.3 - 45.6i)T^{2}$$
73 $$1 + (-6.89 - 5.78i)T + (12.6 + 71.8i)T^{2}$$
79 $$1 + (-7.66 - 6.42i)T + (13.7 + 77.7i)T^{2}$$
83 $$1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (15.4 - 87.6i)T^{2}$$
97 $$1 + (-0.347 + 1.96i)T + (-91.1 - 33.1i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$