# Properties

 Degree $2$ Conductor $3024$ Sign $-0.126 - 0.991i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)7-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (1 + 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (1 − 1.73i)67-s + (−1.5 − 0.866i)73-s + (1 + 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)7-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (1 + 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (1 − 1.73i)67-s + (−1.5 − 0.866i)73-s + (1 + 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $-0.126 - 0.991i$ Motivic weight: $$0$$ Character: $\chi_{3024} (2593, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :0),\ -0.126 - 0.991i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8885928569$$ $$L(\frac12)$$ $$\approx$$ $$0.8885928569$$ $$L(1)$$ 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 - 0.866i)T$$
good5 $$1 + (0.5 - 0.866i)T^{2}$$
11 $$1 + (-0.5 - 0.866i)T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + (0.5 + 0.866i)T^{2}$$
19 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 + (-0.5 + 0.866i)T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}$$
37 $$1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
67 $$1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}$$
79 $$1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (0.5 - 0.866i)T^{2}$$
97 $$1 - 1.73iT - T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.054177946641286646910082250703, −8.362177901967644800640795167687, −7.80051716747258995739023934599, −6.49293798176864093981901029232, −6.33048902036013736782946845390, −5.29546437456800243625744192685, −4.48827454076173295228232081585, −3.45997084599980878527322066409, −2.63365814456375771932375669693, −1.57661444517640536435648763308, 0.53659147058404712339798999000, 2.09677632069388811530758308037, 3.02922798616085085357130631764, 4.20535541113172442382398115999, 4.49439022163802085464188620348, 5.85022465390939629681901195272, 6.44975179771677594213385084891, 7.15165673939126676052198142380, 7.965417739014474246010891126742, 8.648945698951114820520293425504