Properties

Label 2-6008-1.1-c1-0-103
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.40·3-s − 1.83·5-s − 0.970·7-s + 8.58·9-s + 3.13·11-s + 6.37·13-s − 6.24·15-s + 5.07·17-s − 8.17·19-s − 3.30·21-s + 7.55·23-s − 1.63·25-s + 19.0·27-s − 1.60·29-s − 1.14·31-s + 10.6·33-s + 1.78·35-s − 3.51·37-s + 21.7·39-s + 8.67·41-s − 6.25·43-s − 15.7·45-s − 3.02·47-s − 6.05·49-s + 17.2·51-s − 13.9·53-s − 5.74·55-s + ⋯
L(s)  = 1  + 1.96·3-s − 0.820·5-s − 0.366·7-s + 2.86·9-s + 0.944·11-s + 1.76·13-s − 1.61·15-s + 1.23·17-s − 1.87·19-s − 0.720·21-s + 1.57·23-s − 0.326·25-s + 3.65·27-s − 0.298·29-s − 0.205·31-s + 1.85·33-s + 0.301·35-s − 0.577·37-s + 3.47·39-s + 1.35·41-s − 0.954·43-s − 2.34·45-s − 0.441·47-s − 0.865·49-s + 2.42·51-s − 1.91·53-s − 0.774·55-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: \(0\)
Selberg data: \((2,\ 6008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.422803715\)
\(L(\frac12)\) \(\approx\) \(4.422803715\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.40T + 3T^{2} \)
5 \( 1 + 1.83T + 5T^{2} \)
7 \( 1 + 0.970T + 7T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
13 \( 1 - 6.37T + 13T^{2} \)
17 \( 1 - 5.07T + 17T^{2} \)
19 \( 1 + 8.17T + 19T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 + 1.60T + 29T^{2} \)
31 \( 1 + 1.14T + 31T^{2} \)
37 \( 1 + 3.51T + 37T^{2} \)
41 \( 1 - 8.67T + 41T^{2} \)
43 \( 1 + 6.25T + 43T^{2} \)
47 \( 1 + 3.02T + 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
59 \( 1 - 4.93T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 2.18T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 9.76T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.046927313617763824967506137162, −7.77292850514589039340383640274, −6.66969755873321154446289421006, −6.39629792908616543751491582281, −4.87967216802649062309961749868, −3.90375029658441752976404223118, −3.66481251437776392282705150220, −3.07278737223760568714167404761, −1.90251289719132984166403233311, −1.10787907695777920621391368162, 1.10787907695777920621391368162, 1.90251289719132984166403233311, 3.07278737223760568714167404761, 3.66481251437776392282705150220, 3.90375029658441752976404223118, 4.87967216802649062309961749868, 6.39629792908616543751491582281, 6.66969755873321154446289421006, 7.77292850514589039340383640274, 8.046927313617763824967506137162

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