Properties

Label 2-3024-28.27-c1-0-56
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  − 1.73i·5-s + (−0.5 − 2.59i)7-s − 5.19i·11-s − 3.46i·13-s − 3.46i·17-s + 8·19-s + 6.92i·23-s + 2.00·25-s + 6·29-s − 5·31-s + (−4.5 + 0.866i)35-s + 4·37-s − 6.92i·41-s − 3.46i·43-s − 6·47-s + ⋯
L(s)  = 1  − 0.774i·5-s + (−0.188 − 0.981i)7-s − 1.56i·11-s − 0.960i·13-s − 0.840i·17-s + 1.83·19-s + 1.44i·23-s + 0.400·25-s + 1.11·29-s − 0.898·31-s + (−0.760 + 0.146i)35-s + 0.657·37-s − 1.08i·41-s − 0.528i·43-s − 0.875·47-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: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698479425\)
\(L(\frac12)\) \(\approx\) \(1.698479425\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−8.485748344008287363452887878158, −7.56147787022240455212107554339, −7.22293875315507179668820537662, −5.90972994336019980328210711859, −5.43383286617479293384194166205, −4.62065043187181382590848178771, −3.41458956676008800958214186861, −3.09037940669844404205461566467, −1.20458629331792589701431308306, −0.60253261047836479686556714910, 1.58946466492516623393880001500, 2.50456312538850418832409855782, 3.26290870978848290754273136398, 4.47239690492159970291767306484, 5.05005180770035736117198639534, 6.23791272768971002080021076306, 6.64736846153915600103018875532, 7.44158493876838305652500989366, 8.230659355500302233329943368466, 9.154480413383507872776540143350

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