Properties

Degree $2$
Conductor $3024$
Sign $0.755 + 0.654i$
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 − 19-s + 5·25-s + 4·31-s + 37-s − 10.3i·43-s + (5.5 − 4.33i)49-s + 8.66i·61-s − 12.1i·67-s + 1.73i·73-s − 12.1i·79-s + (−1.49 − 4.33i)91-s + 19.0i·97-s + 7·103-s + ⋯
L(s)  = 1  + (0.944 − 0.327i)7-s − 0.480i·13-s − 0.229·19-s + 25-s + 0.718·31-s + 0.164·37-s − 1.58i·43-s + (0.785 − 0.618i)49-s + 1.10i·61-s − 1.48i·67-s + 0.202i·73-s − 1.36i·79-s + (−0.157 − 0.453i)91-s + 1.93i·97-s + 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.755 + 0.654i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.052616582\)
\(L(\frac12)\) \(\approx\) \(2.052616582\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
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

−8.578426893562934369830618639324, −7.911554168247869762044599229253, −7.22706658154691271852652996124, −6.41651793704641959547709427338, −5.45569059392123083395053552887, −4.79353230927762537161910130736, −3.98672837724002051336518278271, −2.95120222167868351862903863597, −1.89869309425830551682810761125, −0.75589049633196307532403972979, 1.11333300124689030659159146722, 2.15677355281794249529549971710, 3.10106985226039559831275240406, 4.31466223597804429417569000636, 4.83376066808061143982723579449, 5.74309944355885487834802716971, 6.55611009572176393682170974254, 7.34151135788103775123796448684, 8.191198838437173175667719206220, 8.666704125815051460347817594089

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