# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.70 − 0.300i)3-s + (−1.26 − 2.19i)5-s + (0.5 − 0.866i)7-s + (2.81 + 1.02i)9-s + (0.233 − 0.405i)11-s + (−2.91 − 5.04i)13-s + (1.5 + 4.12i)15-s + 3.87·17-s + 2.18·19-s + (−1.11 + 1.32i)21-s + (−0.0530 − 0.0918i)23-s + (−0.705 + 1.22i)25-s + (−4.49 − 2.59i)27-s + (−4.39 + 7.60i)29-s + (−3.84 − 6.65i)31-s + ⋯
 L(s)  = 1 + (−0.984 − 0.173i)3-s + (−0.566 − 0.980i)5-s + (0.188 − 0.327i)7-s + (0.939 + 0.342i)9-s + (0.0705 − 0.122i)11-s + (−0.807 − 1.39i)13-s + (0.387 + 1.06i)15-s + 0.940·17-s + 0.501·19-s + (−0.242 + 0.289i)21-s + (−0.0110 − 0.0191i)23-s + (−0.141 + 0.244i)25-s + (−0.866 − 0.499i)27-s + (−0.815 + 1.41i)29-s + (−0.689 − 1.19i)31-s + ⋯

## Functional equation

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

## Invariants

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

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4969853143$$ $$L(\frac12)$$ $$\approx$$ $$0.4969853143$$ $$L(\frac{3}{2})$$ 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 + (1.70 + 0.300i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
good5 $$1 + (1.26 + 2.19i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-0.233 + 0.405i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 - 3.87T + 17T^{2}$$
19 $$1 - 2.18T + 19T^{2}$$
23 $$1 + (0.0530 + 0.0918i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 7.68T + 37T^{2}$$
41 $$1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-0.613 + 1.06i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (2.66 - 4.61i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 0.716T + 53T^{2}$$
59 $$1 + (-0.368 - 0.637i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (0.479 - 0.829i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (4.81 + 8.34i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 13.2T + 71T^{2}$$
73 $$1 + 10.2T + 73T^{2}$$
79 $$1 + (6.31 - 10.9i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (1.36 - 2.36i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + 8.11T + 89T^{2}$$
97 $$1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$