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Properties

Label 2-3024-336.221-c0-0-3
Degree $2$
Conductor $3024$
Sign $-0.997 + 0.0674i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (0.965 − 0.258i)11-s + (−1 − i)13-s + (0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s + (0.366 − 1.36i)19-s − i·22-s + (−0.707 − 1.22i)23-s + (0.866 + 0.5i)25-s + (−1.22 + 0.707i)26-s + 28-s + (−0.707 + 0.707i)29-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (0.965 − 0.258i)11-s + (−1 − i)13-s + (0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s + (0.366 − 1.36i)19-s − i·22-s + (−0.707 − 1.22i)23-s + (0.866 + 0.5i)25-s + (−1.22 + 0.707i)26-s + 28-s + (−0.707 + 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 + 0.0674i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ -0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7594300903\)
\(L(\frac12)\) \(\approx\) \(0.7594300903\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.014998001712147626873018069757, −7.956080335428542904013355501412, −6.85579100082848945555016167147, −6.19714586113599455865678850455, −5.23180908093668136526689525243, −4.65755166215904201794550760210, −3.43190046655435671455206845936, −2.98047148113896644541760576448, −1.95874182055614090015258076880, −0.42586236158610377139009529484, 1.62386898916739422105036172072, 3.24456899527314207463440445892, 3.86823777138271482933620432448, 4.65646229028358778918468963177, 5.55739545236340892535589907238, 6.49649963926358391418308059991, 6.85198835910826815713315120037, 7.58935042773528561455262387640, 8.394618230748233226251745155876, 9.321888248516086961414684212750

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