# Properties

 Degree $2$ Conductor $3024$ Sign $-0.377 + 0.925i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 1.73·5-s + (−1 + 2.44i)7-s − 1.41i·11-s − 2.44i·13-s − 5.19·17-s − 7.34i·19-s + 2.82i·23-s − 2.00·25-s − 7.07i·29-s + 2.44i·31-s + (−1.73 + 4.24i)35-s + 5·37-s − 8.66·41-s − 5·43-s − 8.66·47-s + ⋯
 L(s)  = 1 + 0.774·5-s + (−0.377 + 0.925i)7-s − 0.426i·11-s − 0.679i·13-s − 1.26·17-s − 1.68i·19-s + 0.589i·23-s − 0.400·25-s − 1.31i·29-s + 0.439i·31-s + (−0.292 + 0.717i)35-s + 0.821·37-s − 1.35·41-s − 0.762·43-s − 1.26·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $-0.377 + 0.925i$ Motivic weight: $$1$$ Character: $\chi_{3024} (1889, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :1/2),\ -0.377 + 0.925i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.022352505$$ $$L(\frac12)$$ $$\approx$$ $$1.022352505$$ $$L(\frac{3}{2})$$ 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 + (1 - 2.44i)T$$
good5 $$1 - 1.73T + 5T^{2}$$
11 $$1 + 1.41iT - 11T^{2}$$
13 $$1 + 2.44iT - 13T^{2}$$
17 $$1 + 5.19T + 17T^{2}$$
19 $$1 + 7.34iT - 19T^{2}$$
23 $$1 - 2.82iT - 23T^{2}$$
29 $$1 + 7.07iT - 29T^{2}$$
31 $$1 - 2.44iT - 31T^{2}$$
37 $$1 - 5T + 37T^{2}$$
41 $$1 + 8.66T + 41T^{2}$$
43 $$1 + 5T + 43T^{2}$$
47 $$1 + 8.66T + 47T^{2}$$
53 $$1 + 11.3iT - 53T^{2}$$
59 $$1 - 8.66T + 59T^{2}$$
61 $$1 - 2.44iT - 61T^{2}$$
67 $$1 + 2T + 67T^{2}$$
71 $$1 + 14.1iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 13T + 79T^{2}$$
83 $$1 - 1.73T + 83T^{2}$$
89 $$1 - 10.3T + 89T^{2}$$
97 $$1 - 17.1iT - 97T^{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

−8.587849190724769268193008647750, −7.889975080912016958346346304804, −6.63821226351572235266400003419, −6.36716699825144413890056922871, −5.37230710686915281675145381744, −4.86604173409420923556668248055, −3.55708490797448211037227225228, −2.64485509824527501329535462750, −1.95575135540242145309739374218, −0.29673155227346828015419320557, 1.42506369053523215070259667415, 2.22086211762820082792330300084, 3.48785239393292837442239817350, 4.24918047807647133560683929828, 5.04955768075865872459875128840, 6.13996899913554701368300562726, 6.60589848657121452755309574855, 7.34399508069847759334977751317, 8.240405561723184330601229118515, 9.020628125646532001553119287822