Properties

Label 2-3024-21.20-c1-0-60
Degree $2$
Conductor $3024$
Sign $-0.755 - 0.654i$
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  − 3.46·5-s + (−2 − 1.73i)7-s − 6i·11-s − 1.73i·13-s + 1.73·17-s − 6.92i·19-s + 3i·23-s + 6.99·25-s + 3i·29-s − 5.19i·31-s + (6.92 + 5.99i)35-s − 2·37-s − 6.92·41-s + 11·43-s − 6.92·47-s + ⋯
L(s)  = 1  − 1.54·5-s + (−0.755 − 0.654i)7-s − 1.80i·11-s − 0.480i·13-s + 0.420·17-s − 1.58i·19-s + 0.625i·23-s + 1.39·25-s + 0.557i·29-s − 0.933i·31-s + (1.17 + 1.01i)35-s − 0.328·37-s − 1.08·41-s + 1.67·43-s − 1.01·47-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: $-0.755 - 0.654i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3266624686\)
\(L(\frac12)\) \(\approx\) \(0.3266624686\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 + 3iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−8.176095424448163332753431068417, −7.57684583726571793366998458283, −6.88820994943307952303311548771, −6.06107382667964007208527650944, −5.11427556410438191973106388713, −4.12302560734193529053718358221, −3.38197096344266498248825187001, −2.97237657547156576133852766029, −0.864449789821986888192697866052, −0.13955573195696600379917867544, 1.66021960966188214002536986596, 2.81970290182233458631684285037, 3.79108577758073422401579256900, 4.34127798906834225057164771624, 5.24491206070857153428729584453, 6.31649858095534389427346261971, 7.05116885461731810199961948161, 7.65537171919388584699562550031, 8.347088678130571822000931092976, 9.118404234428819945378087621078

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