Properties

Label 2-6008-1.1-c1-0-138
Degree $2$
Conductor $6008$
Sign $-1$
Analytic cond. $47.9741$
Root an. cond. $6.92633$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28·3-s − 4.29·5-s + 3.51·7-s − 1.34·9-s + 4.71·11-s − 3.44·13-s − 5.52·15-s − 1.72·17-s − 5.09·19-s + 4.52·21-s + 7.19·23-s + 13.4·25-s − 5.59·27-s − 4.72·29-s + 3.97·31-s + 6.07·33-s − 15.0·35-s − 8.45·37-s − 4.44·39-s + 1.87·41-s + 3.39·43-s + 5.75·45-s − 1.38·47-s + 5.34·49-s − 2.22·51-s + 1.09·53-s − 20.2·55-s + ⋯
L(s)  = 1  + 0.743·3-s − 1.91·5-s + 1.32·7-s − 0.446·9-s + 1.42·11-s − 0.956·13-s − 1.42·15-s − 0.419·17-s − 1.16·19-s + 0.987·21-s + 1.50·23-s + 2.68·25-s − 1.07·27-s − 0.877·29-s + 0.713·31-s + 1.05·33-s − 2.54·35-s − 1.39·37-s − 0.711·39-s + 0.292·41-s + 0.517·43-s + 0.857·45-s − 0.202·47-s + 0.763·49-s − 0.311·51-s + 0.150·53-s − 2.73·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-1$
Analytic conductor: \(47.9741\)
Root analytic conductor: \(6.92633\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6008,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 - T \)
good3 \( 1 - 1.28T + 3T^{2} \)
5 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 - 3.51T + 7T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 + 3.44T + 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 + 5.09T + 19T^{2} \)
23 \( 1 - 7.19T + 23T^{2} \)
29 \( 1 + 4.72T + 29T^{2} \)
31 \( 1 - 3.97T + 31T^{2} \)
37 \( 1 + 8.45T + 37T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 3.39T + 43T^{2} \)
47 \( 1 + 1.38T + 47T^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 + 14.6T + 59T^{2} \)
61 \( 1 + 3.40T + 61T^{2} \)
67 \( 1 - 9.01T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 - 7.95T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.990969058116828156701905067311, −7.11850724974645461120803040532, −6.70900936205781044310311751959, −5.30842025907239823591118150897, −4.55329053325840574258733726845, −4.08792165494758252291172210142, −3.34443099070534210296155691524, −2.44085937782819615585385625792, −1.33433715120694600023254292162, 0, 1.33433715120694600023254292162, 2.44085937782819615585385625792, 3.34443099070534210296155691524, 4.08792165494758252291172210142, 4.55329053325840574258733726845, 5.30842025907239823591118150897, 6.70900936205781044310311751959, 7.11850724974645461120803040532, 7.990969058116828156701905067311

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