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Properties

Label 2-3024-63.47-c1-0-35
Degree $2$
Conductor $3024$
Sign $0.784 + 0.620i$
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  + 4.19·5-s + (−1.67 − 2.05i)7-s + 1.66i·11-s + (5.64 − 3.25i)13-s + (−1.45 − 2.52i)17-s + (2.39 + 1.38i)19-s + 2.21i·23-s + 12.6·25-s + (−5.69 − 3.28i)29-s + (−0.414 − 0.239i)31-s + (−7.01 − 8.60i)35-s + (−0.378 + 0.655i)37-s + (0.769 + 1.33i)41-s + (4.79 − 8.31i)43-s + (4.05 + 7.02i)47-s + ⋯
L(s)  = 1  + 1.87·5-s + (−0.631 − 0.775i)7-s + 0.503i·11-s + (1.56 − 0.903i)13-s + (−0.353 − 0.611i)17-s + (0.549 + 0.317i)19-s + 0.462i·23-s + 2.52·25-s + (−1.05 − 0.610i)29-s + (−0.0743 − 0.0429i)31-s + (−1.18 − 1.45i)35-s + (−0.0621 + 0.107i)37-s + (0.120 + 0.208i)41-s + (0.731 − 1.26i)43-s + (0.592 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.784 + 0.620i$
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: \((2,\ 3024,\ (\ :1/2),\ 0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.708364563\)
\(L(\frac12)\) \(\approx\) \(2.708364563\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 2.05i)T \)
good5 \( 1 - 4.19T + 5T^{2} \)
11 \( 1 - 1.66iT - 11T^{2} \)
13 \( 1 + (-5.64 + 3.25i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.45 + 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.39 - 1.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.21iT - 23T^{2} \)
29 \( 1 + (5.69 + 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.414 + 0.239i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.378 - 0.655i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.769 - 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.79 + 8.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.05 - 7.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.11 - 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.426 + 0.739i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.89 + 2.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.69 + 13.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.89iT - 71T^{2} \)
73 \( 1 + (6.22 - 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.52 + 9.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.162 + 0.280i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.86 + 4.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.22 - 2.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.930298649362865248156079965095, −7.79645116828022371019518549150, −7.05172593548391588504877345055, −6.13223095296470584872806358089, −5.84595533246114528816509308720, −4.93953578807285500928226903855, −3.78308804447526878495371543954, −2.94246330568907598754636148221, −1.87465263522720269813751808924, −0.929828842417082119139616752310, 1.29014499316218762240556934903, 2.12624527508441894252162452003, 3.00857540009163892275669885886, 4.01755106015452840303111024379, 5.31881569280237470607360078070, 5.81631373121001086496833114989, 6.38104862539434411153070780228, 6.94249468391711701728502259247, 8.454961226861335013131005627174, 8.906161520336742465632900944911

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