Properties

Degree $2$
Conductor $3024$
Sign $0.327 - 0.944i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 0.866i)7-s − 1.73i·13-s + 8.66i·19-s − 5·25-s + 10.3i·31-s + 37-s + 8·43-s + (5.5 + 4.33i)49-s − 8.66i·61-s − 11·67-s + 1.73i·73-s + 13·79-s + (1.49 − 4.33i)91-s + 19.0i·97-s + 19.0i·103-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)7-s − 0.480i·13-s + 1.98i·19-s − 25-s + 1.86i·31-s + 0.164·37-s + 1.21·43-s + (0.785 + 0.618i)49-s − 1.10i·61-s − 1.34·67-s + 0.202i·73-s + 1.46·79-s + (0.157 − 0.453i)91-s + 1.93i·97-s + 1.87i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.327 - 0.944i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800930228\)
\(L(\frac12)\) \(\approx\) \(1.800930228\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + 11T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−8.785551217603960942057289048301, −7.953148327174725682167949539983, −7.70943260864157341491705331253, −6.52636116670720468278558279275, −5.72060992980244575548564254827, −5.13976238536448664891883172980, −4.15602799619718188719085961606, −3.33158839020362553575561736016, −2.14429156638193104562435287295, −1.27094050615740478147707588121, 0.59279181459941608195641281408, 1.89531933809183553671021224480, 2.74781668668536905373068115009, 4.09572157708754429155199017091, 4.53691187361494717572506307573, 5.48237915468850255873683290519, 6.29596164257119100121439676028, 7.28539439181109589120339752562, 7.67101250455019388651870754861, 8.632135251851938984692555237765

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