Properties

Label 2-6008-1.1-c1-0-139
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  + 2.74·3-s + 2.04·5-s + 1.49·7-s + 4.53·9-s + 2.37·11-s + 2.66·13-s + 5.60·15-s + 0.879·17-s + 6.71·19-s + 4.11·21-s − 3.16·23-s − 0.831·25-s + 4.20·27-s + 1.61·29-s − 5.94·31-s + 6.51·33-s + 3.06·35-s − 8.58·37-s + 7.30·39-s + 10.7·41-s − 4.67·43-s + 9.25·45-s + 5.49·47-s − 4.75·49-s + 2.41·51-s − 7.37·53-s + 4.84·55-s + ⋯
L(s)  = 1  + 1.58·3-s + 0.913·5-s + 0.566·7-s + 1.51·9-s + 0.716·11-s + 0.738·13-s + 1.44·15-s + 0.213·17-s + 1.54·19-s + 0.897·21-s − 0.659·23-s − 0.166·25-s + 0.808·27-s + 0.300·29-s − 1.06·31-s + 1.13·33-s + 0.517·35-s − 1.41·37-s + 1.16·39-s + 1.67·41-s − 0.713·43-s + 1.37·45-s + 0.801·47-s − 0.678·49-s + 0.338·51-s − 1.01·53-s + 0.653·55-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

\(L(1)\) \(\approx\) \(5.368840611\)
\(L(\frac12)\) \(\approx\) \(5.368840611\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 3T^{2} \)
5 \( 1 - 2.04T + 5T^{2} \)
7 \( 1 - 1.49T + 7T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 - 2.66T + 13T^{2} \)
17 \( 1 - 0.879T + 17T^{2} \)
19 \( 1 - 6.71T + 19T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 - 1.61T + 29T^{2} \)
31 \( 1 + 5.94T + 31T^{2} \)
37 \( 1 + 8.58T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 4.67T + 43T^{2} \)
47 \( 1 - 5.49T + 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 - 9.22T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + 4.92T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 5.17T + 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 + 1.18T + 83T^{2} \)
89 \( 1 + 7.62T + 89T^{2} \)
97 \( 1 + 1.00T + 97T^{2} \)
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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

−8.111241711609890049999992123286, −7.57566802629900758532471001172, −6.83536297266988482102750112571, −5.89402825807477481320381535528, −5.27429345213263463903958297980, −4.15667756630638624412384953067, −3.55369245847305201209183941474, −2.77992399727208271753661827833, −1.82287330156070356077130239307, −1.33847927448735934667822804687, 1.33847927448735934667822804687, 1.82287330156070356077130239307, 2.77992399727208271753661827833, 3.55369245847305201209183941474, 4.15667756630638624412384953067, 5.27429345213263463903958297980, 5.89402825807477481320381535528, 6.83536297266988482102750112571, 7.57566802629900758532471001172, 8.111241711609890049999992123286

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