# Properties

 Degree $2$ Conductor $3024$ Sign $0.821 + 0.570i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.841 − 1.45i)5-s + (−1.65 + 2.06i)7-s + (−0.622 − 1.07i)11-s + (1.96 + 3.39i)13-s + (1.62 − 2.81i)17-s + (−2.36 − 4.09i)19-s + (0.199 − 0.344i)23-s + (1.08 + 1.87i)25-s + (3.19 − 5.54i)29-s + 0.578·31-s + (1.61 + 4.14i)35-s + (2.72 + 4.71i)37-s + (−4.20 − 7.27i)41-s + (−2.46 + 4.26i)43-s − 0.425·47-s + ⋯
 L(s)  = 1 + (0.376 − 0.651i)5-s + (−0.625 + 0.780i)7-s + (−0.187 − 0.325i)11-s + (0.543 + 0.941i)13-s + (0.394 − 0.683i)17-s + (−0.541 − 0.938i)19-s + (0.0415 − 0.0718i)23-s + (0.216 + 0.375i)25-s + (0.594 − 1.02i)29-s + 0.103·31-s + (0.273 + 0.701i)35-s + (0.447 + 0.774i)37-s + (−0.656 − 1.13i)41-s + (−0.375 + 0.650i)43-s − 0.0620·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $0.821 + 0.570i$ Motivic weight: $$1$$ Character: $\chi_{3024} (2881, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :1/2),\ 0.821 + 0.570i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.761110949$$ $$L(\frac12)$$ $$\approx$$ $$1.761110949$$ $$L(\frac{3}{2})$$ 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 + (1.65 - 2.06i)T$$
good5 $$1 + (-0.841 + 1.45i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (0.622 + 1.07i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-0.199 + 0.344i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-3.19 + 5.54i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 0.578T + 31T^{2}$$
37 $$1 + (-2.72 - 4.71i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (4.20 + 7.27i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (2.46 - 4.26i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 0.425T + 47T^{2}$$
53 $$1 + (-0.466 + 0.807i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 6.05T + 59T^{2}$$
61 $$1 - 10.2T + 61T^{2}$$
67 $$1 - 9.41T + 67T^{2}$$
71 $$1 - 8.46T + 71T^{2}$$
73 $$1 + (-6.82 + 11.8i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 - 5.53T + 79T^{2}$$
83 $$1 + (8.03 - 13.9i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (-6.03 - 10.4i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)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

−8.628317788718120007347907323948, −8.213368988907090558903517256249, −6.88382279278991967690795095268, −6.46806836711530400555663618066, −5.50937679731944989020605522901, −4.93920004839525798124516015730, −3.92761323715287308960034876335, −2.88601056113741497118487344950, −2.03308433799188136229684796261, −0.69351962845897691422040194358, 0.957606770476427571434360764802, 2.23801183385083374500013323349, 3.30841242458757638951994098681, 3.83173388331015050974927654251, 4.98680236979219005615160673331, 5.94001873669171304345004780330, 6.50001479419567640904914378395, 7.21151890096084207975062039611, 8.063982577819911947207403597431, 8.650635573293036320314445384216