# Properties

 Degree $2$ Conductor $1008$ Sign $-0.635 - 0.771i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + 1.41i)3-s + (−1.72 − 2.98i)5-s + (−0.5 + 0.866i)7-s + (−1.00 − 2.82i)9-s + (1 − 1.73i)11-s + (2.44 + 4.24i)13-s + (5.94 + 0.548i)15-s + 2·17-s − 7.44·19-s + (−0.724 − 1.57i)21-s + (−0.5 − 0.866i)23-s + (−3.44 + 5.97i)25-s + (5.00 + 1.41i)27-s + (−1.44 + 2.51i)29-s + (3 + 5.19i)31-s + ⋯
 L(s)  = 1 + (−0.577 + 0.816i)3-s + (−0.771 − 1.33i)5-s + (−0.188 + 0.327i)7-s + (−0.333 − 0.942i)9-s + (0.301 − 0.522i)11-s + (0.679 + 1.17i)13-s + (1.53 + 0.141i)15-s + 0.485·17-s − 1.70·19-s + (−0.158 − 0.343i)21-s + (−0.104 − 0.180i)23-s + (−0.689 + 1.19i)25-s + (0.962 + 0.272i)27-s + (−0.269 + 0.466i)29-s + (0.538 + 0.933i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $-0.635 - 0.771i$ Motivic weight: $$1$$ Character: $\chi_{1008} (673, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1008,\ (\ :1/2),\ -0.635 - 0.771i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4837140750$$ $$L(\frac12)$$ $$\approx$$ $$0.4837140750$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (1 - 1.41i)T$$
7 $$1 + (0.5 - 0.866i)T$$
good5 $$1 + (1.72 + 2.98i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 - 2T + 17T^{2}$$
19 $$1 + 7.44T + 19T^{2}$$
23 $$1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (1.44 - 2.51i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 7.79T + 37T^{2}$$
41 $$1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (1.44 - 2.51i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 1.10T + 53T^{2}$$
59 $$1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (5.72 - 9.91i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 9.89T + 71T^{2}$$
73 $$1 - 2.89T + 73T^{2}$$
79 $$1 + (-3.94 + 6.84i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + 7.10T + 89T^{2}$$
97 $$1 + (-3.44 + 5.97i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$