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Properties

Label 2-3024-63.20-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.999 - 0.0232i$
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  + (0.895 + 1.55i)5-s + (2.30 − 1.30i)7-s + (−2.07 − 1.20i)11-s + (−4.23 + 2.44i)13-s − 3.66·17-s − 3.01i·19-s + (3.26 − 1.88i)23-s + (0.897 − 1.55i)25-s + (5.68 + 3.28i)29-s + (4.02 − 2.32i)31-s + (4.08 + 2.40i)35-s + 9.36·37-s + (4.04 + 6.99i)41-s + (3.48 − 6.02i)43-s + (2.56 − 4.44i)47-s + ⋯
L(s)  = 1  + (0.400 + 0.693i)5-s + (0.870 − 0.492i)7-s + (−0.627 − 0.362i)11-s + (−1.17 + 0.678i)13-s − 0.888·17-s − 0.692i·19-s + (0.680 − 0.392i)23-s + (0.179 − 0.310i)25-s + (1.05 + 0.609i)29-s + (0.722 − 0.417i)31-s + (0.690 + 0.405i)35-s + 1.53·37-s + (0.631 + 1.09i)41-s + (0.530 − 0.919i)43-s + (0.374 − 0.648i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.999 - 0.0232i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.999 - 0.0232i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.029121809\)
\(L(\frac12)\) \(\approx\) \(2.029121809\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 + 1.30i)T \)
good5 \( 1 + (-0.895 - 1.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.07 + 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.23 - 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 + 3.01iT - 19T^{2} \)
23 \( 1 + (-3.26 + 1.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.68 - 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.02 + 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (-4.04 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.48 + 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-7.29 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.81 - 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.285 - 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (-1.51 + 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.00 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + (-4.77 - 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.675959097793051867593051302294, −7.963449769860728352159196396555, −7.03947133089329791881419000465, −6.73613069903830153022098613936, −5.63900433778622113079969409445, −4.71151052452835651664963475784, −4.27152261953106031069278765299, −2.67631088317178371556384455566, −2.41056768159000686451245808894, −0.838194328422581196328556067859, 0.913536091751455614350517560345, 2.14956284193353438639065977019, 2.78227122983566534525918041641, 4.29450303617897280817106729307, 4.96739166709554569121364279671, 5.42400463310872333635118855261, 6.36075331509112239426922974232, 7.43356546388726916458286453352, 7.995871059739034020484473273556, 8.660841846407851938390455903660

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