# Properties

 Degree $2$ Conductor $1008$ Sign $0.0633 - 0.997i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3 + 1.73i)5-s + (0.5 − 2.59i)7-s + (3 + 1.73i)11-s − 1.73i·13-s + (2.5 + 4.33i)19-s + (−6 + 3.46i)23-s + (3.5 − 6.06i)25-s + (−2.5 + 4.33i)31-s + (3 + 8.66i)35-s + (5.5 + 9.52i)37-s + 3.46i·41-s + 8.66i·43-s + (3 + 5.19i)47-s + (−6.5 − 2.59i)49-s + (6 − 10.3i)53-s + ⋯
 L(s)  = 1 + (−1.34 + 0.774i)5-s + (0.188 − 0.981i)7-s + (0.904 + 0.522i)11-s − 0.480i·13-s + (0.573 + 0.993i)19-s + (−1.25 + 0.722i)23-s + (0.700 − 1.21i)25-s + (−0.449 + 0.777i)31-s + (0.507 + 1.46i)35-s + (0.904 + 1.56i)37-s + 0.541i·41-s + 1.32i·43-s + (0.437 + 0.757i)47-s + (−0.928 − 0.371i)49-s + (0.824 − 1.42i)53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $0.0633 - 0.997i$ Motivic weight: $$1$$ Character: $\chi_{1008} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1008,\ (\ :1/2),\ 0.0633 - 0.997i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.012069709$$ $$L(\frac12)$$ $$\approx$$ $$1.012069709$$ $$L(\frac{3}{2})$$ 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 + (-0.5 + 2.59i)T$$
good5 $$1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + 1.73iT - 13T^{2}$$
17 $$1 + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (6 - 3.46i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 3.46iT - 41T^{2}$$
43 $$1 - 8.66iT - 43T^{2}$$
47 $$1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 3.46iT - 71T^{2}$$
73 $$1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 - 18T + 83T^{2}$$
89 $$1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 - 6.92iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$