Properties

Degree $2$
Conductor $3024$
Sign $0.5 - 0.866i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46i·5-s − 2.64·7-s + 4.58i·13-s − 1.73i·17-s − 5.29·19-s + 4.58i·23-s − 6.99·25-s + 7.93·29-s − 2.64·31-s + 9.16i·35-s + 4·37-s + 6.92i·41-s − 1.73i·43-s − 6·47-s + 7.00·49-s + ⋯
L(s)  = 1  − 1.54i·5-s − 0.999·7-s + 1.27i·13-s − 0.420i·17-s − 1.21·19-s + 0.955i·23-s − 1.39·25-s + 1.47·29-s − 0.475·31-s + 1.54i·35-s + 0.657·37-s + 1.08i·41-s − 0.264i·43-s − 0.875·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.5 - 0.866i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8776394971\)
\(L(\frac12)\) \(\approx\) \(0.8776394971\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 - 4.58iT - 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 9.16iT - 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 - 4.58iT - 71T^{2} \)
73 \( 1 + 9.16iT - 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 5.19iT - 89T^{2} \)
97 \( 1 + 9.16iT - 97T^{2} \)
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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

−8.879374791507878557709670929902, −8.335623516144387085833434931604, −7.33935125088105743776056311064, −6.47870380945965778313310664847, −5.88057113563610627400037320937, −4.76335103094274571025763720278, −4.38317100606364102124423469455, −3.33700483409131087015183657450, −2.10826812791739662003575078402, −1.02634958463653788423674866876, 0.30996924983463594574558091552, 2.21486702787989740790102378284, 2.99692983859679776130038916266, 3.54581441697250822623071516613, 4.63215516681622433331001492695, 5.87370656747971866759144125921, 6.42716797466483306966862560216, 6.86683169056934973055324779701, 7.84670087808327420889141006231, 8.455098356724067382512696698516

Graph of the $Z$-function along the critical line