# Properties

 Degree $2$ Conductor $1008$ Sign $0.991 - 0.126i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.5 − 0.866i)5-s + 2.64·7-s + (3.96 + 2.29i)11-s + 3.46i·13-s + (−4.5 − 2.59i)17-s + (1.32 + 2.29i)19-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s + (−1.32 + 2.29i)31-s + (3.96 − 2.29i)35-s + (−3.5 − 6.06i)37-s + 3.46i·41-s − 9.16i·43-s + (3.96 + 6.87i)47-s + 7.00·49-s + ⋯
 L(s)  = 1 + (0.670 − 0.387i)5-s + 0.999·7-s + (1.19 + 0.690i)11-s + 0.960i·13-s + (−1.09 − 0.630i)17-s + (0.303 + 0.525i)19-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s + (−0.237 + 0.411i)31-s + (0.670 − 0.387i)35-s + (−0.575 − 0.996i)37-s + 0.541i·41-s − 1.39i·43-s + (0.578 + 1.00i)47-s + 49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $0.991 - 0.126i$ Motivic weight: $$1$$ Character: $\chi_{1008} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1008,\ (\ :1/2),\ 0.991 - 0.126i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.115665273$$ $$L(\frac12)$$ $$\approx$$ $$2.115665273$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 - 2.64T$$
good5 $$1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-3.96 - 2.29i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 - 3.46iT - 13T^{2}$$
17 $$1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 3.46iT - 41T^{2}$$
43 $$1 + 9.16iT - 43T^{2}$$
47 $$1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 - 9.16iT - 71T^{2}$$
73 $$1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (3.96 - 2.29i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + 3.46iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$