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Properties

Label 2-3024-63.16-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.621 - 0.783i$
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  + 0.0619·5-s + (1.63 − 2.07i)7-s − 3.18·11-s + (−0.252 − 0.437i)13-s + (0.554 + 0.960i)17-s + (−0.933 + 1.61i)19-s − 6.20·23-s − 4.99·25-s + (−2.39 + 4.15i)29-s + (−1.26 + 2.19i)31-s + (0.101 − 0.128i)35-s + (−4.26 + 7.38i)37-s + (4.94 + 8.56i)41-s + (3.95 − 6.85i)43-s + (−3.29 − 5.70i)47-s + ⋯
L(s)  = 1  + 0.0277·5-s + (0.618 − 0.785i)7-s − 0.958·11-s + (−0.0700 − 0.121i)13-s + (0.134 + 0.233i)17-s + (−0.214 + 0.370i)19-s − 1.29·23-s − 0.999·25-s + (−0.445 + 0.770i)29-s + (−0.227 + 0.394i)31-s + (0.0171 − 0.0217i)35-s + (−0.700 + 1.21i)37-s + (0.772 + 1.33i)41-s + (0.603 − 1.04i)43-s + (−0.480 − 0.831i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5631907925\)
\(L(\frac12)\) \(\approx\) \(0.5631907925\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 2.07i)T \)
good5 \( 1 - 0.0619T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + (0.252 + 0.437i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.554 - 0.960i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.933 - 1.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.20T + 23T^{2} \)
29 \( 1 + (2.39 - 4.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.26 - 2.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 - 7.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.94 - 8.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.95 + 6.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.29 + 5.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.58 - 2.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.50 - 7.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.94 - 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 - 3.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.48 + 2.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.17 + 3.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.30 + 7.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.859098040924999239905143410524, −8.054707573364911039937975423783, −7.67573923075800718512813916204, −6.83256869818640681712896976329, −5.84739012548405181660887118393, −5.18186370484878180458796471517, −4.27022764454945584899240491088, −3.54376106346066179298467621919, −2.35353197121692136123180560096, −1.36431584702309107628910382218, 0.16691512613580956005715200130, 1.94924566924866878505792627238, 2.47592529976659381951447581315, 3.74187221704104714795486959405, 4.60096538516546676673877654478, 5.54175980388129165862069021963, 5.90820322291537070320619516209, 7.04382661372079214921032994733, 8.029548248834627688766408844687, 8.116055080318452261803081069589

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