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Properties

Label 2-3024-63.4-c1-0-41
Degree $2$
Conductor $3024$
Sign $-0.866 + 0.499i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·5-s + (1.78 − 1.95i)7-s + 0.399·11-s + (1.44 − 2.49i)13-s + (0.176 − 0.305i)17-s + (−2.84 − 4.93i)19-s − 0.877·23-s − 0.571·25-s + (−0.874 − 1.51i)29-s + (4.56 + 7.91i)31-s + (−3.75 + 4.11i)35-s + (−3.39 − 5.88i)37-s + (−1.20 + 2.08i)41-s + (−0.276 − 0.479i)43-s + (−5.86 + 10.1i)47-s + ⋯
L(s)  = 1  − 0.941·5-s + (0.674 − 0.738i)7-s + 0.120·11-s + (0.400 − 0.693i)13-s + (0.0428 − 0.0741i)17-s + (−0.653 − 1.13i)19-s − 0.182·23-s − 0.114·25-s + (−0.162 − 0.281i)29-s + (0.820 + 1.42i)31-s + (−0.634 + 0.694i)35-s + (−0.558 − 0.966i)37-s + (−0.187 + 0.325i)41-s + (−0.0422 − 0.0730i)43-s + (−0.856 + 1.48i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.866 + 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7819691852\)
\(L(\frac12)\) \(\approx\) \(0.7819691852\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
good5 \( 1 + 2.10T + 5T^{2} \)
11 \( 1 - 0.399T + 11T^{2} \)
13 \( 1 + (-1.44 + 2.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.176 + 0.305i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 + 4.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.877T + 23T^{2} \)
29 \( 1 + (0.874 + 1.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.56 - 7.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.20 - 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.276 + 0.479i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.86 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.07 + 3.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.66 - 8.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.03 + 8.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.601 - 1.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 + 0.546i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.24 - 2.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.59 + 7.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.84 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.380992647762673326972719737116, −7.65515570822879961726883064796, −7.09122873277971318145287673307, −6.23385503604862628034105759341, −5.13986690823616621624466121231, −4.44481946907374066706182372422, −3.74263651360395599497003657887, −2.80965161390714620059856404817, −1.42983808115646530682216129754, −0.25413043930684594684124467622, 1.46979707702728331505364263944, 2.42428817097592374590534293254, 3.70815057814607453141797787776, 4.20556657267568776886439495937, 5.17262960946257382135504035246, 6.01392484770473138226993356195, 6.77151947302667293663318458755, 7.75197586413963123418704520701, 8.290827063178528773239984819493, 8.768654929330574893942311193569

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