# Properties

 Label 2-3072-8.5-c1-0-36 Degree $2$ Conductor $3072$ Sign $1$ Analytic cond. $24.5300$ Root an. cond. $4.95278$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + i·3-s + 2.47i·5-s − 2.55·7-s − 9-s + 0.669i·11-s − 4.08i·13-s − 2.47·15-s + 6.44·17-s − 6.44i·19-s − 2.55i·21-s + 2.82·23-s − 1.11·25-s − i·27-s − 4.35i·29-s − 6.55·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.10i·5-s − 0.966·7-s − 0.333·9-s + 0.201i·11-s − 1.13i·13-s − 0.638·15-s + 1.56·17-s − 1.47i·19-s − 0.558i·21-s + 0.589·23-s − 0.223·25-s − 0.192i·27-s − 0.808i·29-s − 1.17·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3072$$    =    $$2^{10} \cdot 3$$ Sign: $1$ Analytic conductor: $$24.5300$$ Root analytic conductor: $$4.95278$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3072} (1537, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3072,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.427841727$$ $$L(\frac12)$$ $$\approx$$ $$1.427841727$$ $$L(\frac{3}{2})$$ 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 - iT$$
good5 $$1 - 2.47iT - 5T^{2}$$
7 $$1 + 2.55T + 7T^{2}$$
11 $$1 - 0.669iT - 11T^{2}$$
13 $$1 + 4.08iT - 13T^{2}$$
17 $$1 - 6.44T + 17T^{2}$$
19 $$1 + 6.44iT - 19T^{2}$$
23 $$1 - 2.82T + 23T^{2}$$
29 $$1 + 4.35iT - 29T^{2}$$
31 $$1 + 6.55T + 31T^{2}$$
37 $$1 + 3.85iT - 37T^{2}$$
41 $$1 + 0.788T + 41T^{2}$$
43 $$1 + 0.550iT - 43T^{2}$$
47 $$1 - 2.82T + 47T^{2}$$
53 $$1 - 3.64iT - 53T^{2}$$
59 $$1 + 5.65iT - 59T^{2}$$
61 $$1 + 6.20iT - 61T^{2}$$
67 $$1 - 2.99iT - 67T^{2}$$
71 $$1 - 5.11T + 71T^{2}$$
73 $$1 - 14.7T + 73T^{2}$$
79 $$1 + 6.31T + 79T^{2}$$
83 $$1 - 0.907iT - 83T^{2}$$
89 $$1 - 6.31T + 89T^{2}$$
97 $$1 - 12.6T + 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

−8.864044449426771805751739969976, −7.80143472639530778533690494164, −7.21104565495802094867417739106, −6.45344605186764234458029706025, −5.66690120220221941689534838957, −4.94839457235190040967003709317, −3.64390511414266871370337933563, −3.19439322986092907175846329698, −2.46416384824110385239545507976, −0.55100458888902263295292373052, 0.976980473448459421816177034803, 1.79164628328548991378238026534, 3.19171548592230284062624845366, 3.85470015373742224190889269354, 5.01300175062752681504219918948, 5.67879887776666989884640320044, 6.43112059276035337046748844864, 7.22731956880204589860313973094, 7.980493157110369153245819827599, 8.744884202914717085420417432723