# Properties

 Degree $2$ Conductor $3024$ Sign $-0.999 + 0.0286i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.234 − 0.405i)5-s + (−0.212 + 2.63i)7-s + (0.674 − 1.16i)11-s + (−3.16 + 5.48i)13-s + (2.47 + 4.28i)17-s + (−2.38 + 4.13i)19-s + (−3.81 − 6.60i)23-s + (2.39 − 4.14i)25-s + (1.80 + 3.12i)29-s − 6.49·31-s + (1.11 − 0.531i)35-s + (5.24 − 9.07i)37-s + (0.0251 − 0.0435i)41-s + (0.431 + 0.748i)43-s − 10.9·47-s + ⋯
 L(s)  = 1 + (−0.104 − 0.181i)5-s + (−0.0802 + 0.996i)7-s + (0.203 − 0.352i)11-s + (−0.877 + 1.52i)13-s + (0.599 + 1.03i)17-s + (−0.548 + 0.949i)19-s + (−0.795 − 1.37i)23-s + (0.478 − 0.828i)25-s + (0.335 + 0.580i)29-s − 1.16·31-s + (0.189 − 0.0897i)35-s + (0.861 − 1.49i)37-s + (0.00392 − 0.00680i)41-s + (0.0658 + 0.114i)43-s − 1.60·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $-0.999 + 0.0286i$ Motivic weight: $$1$$ Character: $\chi_{3024} (2305, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :1/2),\ -0.999 + 0.0286i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4148849414$$ $$L(\frac12)$$ $$\approx$$ $$0.4148849414$$ $$L(\frac{3}{2})$$ 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 + (0.212 - 2.63i)T$$
good5 $$1 + (0.234 + 0.405i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-0.674 + 1.16i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (3.16 - 5.48i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-2.47 - 4.28i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3.81 + 6.60i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.80 - 3.12i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 6.49T + 31T^{2}$$
37 $$1 + (-5.24 + 9.07i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-0.0251 + 0.0435i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.431 - 0.748i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 10.9T + 47T^{2}$$
53 $$1 + (5.84 + 10.1i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 3.87T + 59T^{2}$$
61 $$1 - 3.74T + 61T^{2}$$
67 $$1 - 2.64T + 67T^{2}$$
71 $$1 + 7.04T + 71T^{2}$$
73 $$1 + (3.30 + 5.71i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + 3.17T + 79T^{2}$$
83 $$1 + (-4.90 - 8.49i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (5.30 - 9.19i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-6.97 - 12.0i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$