Properties

Degree $2$
Conductor $3024$
Sign $0.391 - 0.920i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.712 − 1.23i)5-s + (2.36 + 1.19i)7-s + (2.46 + 4.27i)11-s + (−1.37 − 2.38i)13-s + (−0.559 + 0.969i)17-s + (2.00 + 3.47i)19-s + (−2.71 + 4.70i)23-s + (1.48 + 2.57i)25-s + (−3.40 + 5.89i)29-s − 2.50·31-s + (3.15 − 2.06i)35-s + (0.709 + 1.22i)37-s + (−0.124 − 0.215i)41-s + (0.498 − 0.863i)43-s − 9.47·47-s + ⋯
L(s)  = 1  + (0.318 − 0.551i)5-s + (0.892 + 0.451i)7-s + (0.743 + 1.28i)11-s + (−0.381 − 0.661i)13-s + (−0.135 + 0.235i)17-s + (0.460 + 0.797i)19-s + (−0.566 + 0.981i)23-s + (0.296 + 0.514i)25-s + (−0.632 + 1.09i)29-s − 0.450·31-s + (0.533 − 0.348i)35-s + (0.116 + 0.202i)37-s + (−0.0194 − 0.0336i)41-s + (0.0759 − 0.131i)43-s − 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.391 - 0.920i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006528217\)
\(L(\frac12)\) \(\approx\) \(2.006528217\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 - 1.19i)T \)
good5 \( 1 + (-0.712 + 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.46 - 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.37 + 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.559 - 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.40 - 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.124 + 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.498 + 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + (-0.410 + 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 0.0752T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.76 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 + 4.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.961754974511951920924445402852, −8.034821493538708699685243456234, −7.51628550386104017385116397759, −6.64434591762537254911773110939, −5.51424243533291368275724228644, −5.18032458998077463183256893438, −4.27757187397875415319627991160, −3.31232852500837920207624014562, −1.91178267895621716900892122160, −1.44859730147662817735110059670, 0.63135890695479856478543374021, 1.90574999421930746477905986473, 2.83947015808897309819118902984, 3.91107742814149410555035423141, 4.61424435522477046089365920018, 5.57241247496281633646740836557, 6.44355506231749254201673363308, 6.94821978216258596193968874074, 7.903619172270740214318486678561, 8.527696950597654041334164896898

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