Properties

Label 2-6008-1.1-c1-0-137
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  + 0.495·3-s − 1.65·5-s + 2.30·7-s − 2.75·9-s − 1.40·11-s − 1.15·13-s − 0.821·15-s + 5.74·17-s + 2.37·19-s + 1.14·21-s + 1.50·23-s − 2.25·25-s − 2.85·27-s − 6.31·29-s − 4.08·31-s − 0.695·33-s − 3.81·35-s + 2.99·37-s − 0.572·39-s + 11.4·41-s + 0.314·43-s + 4.56·45-s − 2.19·47-s − 1.68·49-s + 2.84·51-s − 5.73·53-s + 2.32·55-s + ⋯
L(s)  = 1  + 0.286·3-s − 0.740·5-s + 0.871·7-s − 0.918·9-s − 0.423·11-s − 0.320·13-s − 0.212·15-s + 1.39·17-s + 0.545·19-s + 0.249·21-s + 0.313·23-s − 0.451·25-s − 0.549·27-s − 1.17·29-s − 0.734·31-s − 0.121·33-s − 0.645·35-s + 0.493·37-s − 0.0916·39-s + 1.79·41-s + 0.0478·43-s + 0.680·45-s − 0.319·47-s − 0.240·49-s + 0.398·51-s − 0.787·53-s + 0.313·55-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 \( 1 - 0.495T + 3T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 - 2.30T + 7T^{2} \)
11 \( 1 + 1.40T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 - 5.74T + 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 - 1.50T + 23T^{2} \)
29 \( 1 + 6.31T + 29T^{2} \)
31 \( 1 + 4.08T + 31T^{2} \)
37 \( 1 - 2.99T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 0.314T + 43T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 + 5.43T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 - 4.14T + 89T^{2} \)
97 \( 1 + 7.53T + 97T^{2} \)
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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

−7.80144970282478068284126831231, −7.46738164050604122906076967768, −6.18193030092302871127922607108, −5.47255071365318586808797533697, −4.93157500483256326617822893608, −3.92793543605960891003156535862, −3.26219759510693516539041014090, −2.42876621797719987144078224602, −1.32159012561950537040381743680, 0, 1.32159012561950537040381743680, 2.42876621797719987144078224602, 3.26219759510693516539041014090, 3.92793543605960891003156535862, 4.93157500483256326617822893608, 5.47255071365318586808797533697, 6.18193030092302871127922607108, 7.46738164050604122906076967768, 7.80144970282478068284126831231

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