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Properties

Label 2-3024-63.47-c1-0-6
Degree $2$
Conductor $3024$
Sign $-0.150 - 0.988i$
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  + 0.0525·5-s + (−2.44 − 1.01i)7-s + 2.48i·11-s + (2.51 − 1.45i)13-s + (−2.88 − 4.99i)17-s + (2.92 + 1.69i)19-s + 8.63i·23-s − 4.99·25-s + (6.23 + 3.60i)29-s + (−8.59 − 4.96i)31-s + (−0.128 − 0.0535i)35-s + (−0.770 + 1.33i)37-s + (−0.392 − 0.679i)41-s + (2.03 − 3.51i)43-s + (0.657 + 1.13i)47-s + ⋯
L(s)  = 1  + 0.0235·5-s + (−0.922 − 0.385i)7-s + 0.749i·11-s + (0.698 − 0.403i)13-s + (−0.700 − 1.21i)17-s + (0.671 + 0.387i)19-s + 1.79i·23-s − 0.999·25-s + (1.15 + 0.668i)29-s + (−1.54 − 0.890i)31-s + (−0.0216 − 0.00905i)35-s + (−0.126 + 0.219i)37-s + (−0.0612 − 0.106i)41-s + (0.309 − 0.536i)43-s + (0.0959 + 0.166i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9527253803\)
\(L(\frac12)\) \(\approx\) \(0.9527253803\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.01i)T \)
good5 \( 1 - 0.0525T + 5T^{2} \)
11 \( 1 - 2.48iT - 11T^{2} \)
13 \( 1 + (-2.51 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.88 + 4.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 1.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.63iT - 23T^{2} \)
29 \( 1 + (-6.23 - 3.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.59 + 4.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.770 - 1.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.392 + 0.679i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.03 + 3.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.657 - 1.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.710 - 0.410i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.32 - 4.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.87 + 2.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.95 - 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.51iT - 71T^{2} \)
73 \( 1 + (-6.75 + 3.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.50 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.14 - 5.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.93 - 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.57 - 0.909i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.238059429561798371490926187459, −8.059244545933102607592569776608, −7.30601014867902429438836614538, −6.85228182364963073498249047265, −5.81237548243895854873279111927, −5.22654568097940478318823831226, −4.04469658755903758028490678705, −3.45095801223002693281499563938, −2.42157639211092803928430263233, −1.15329855464029576110486930811, 0.31941570802802713911209248368, 1.82034834825440780328909800357, 2.89385419290298872404090818422, 3.69407613365730249127785809118, 4.52268141085341261815985902532, 5.67172373018832281093667440201, 6.29704829608667337784844460574, 6.75066816381599122377036416476, 7.900152446192634744314304229818, 8.693027481628192462279320042086

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