Properties

Label 2-6008-1.1-c1-0-114
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  − 2.71·3-s + 1.94·5-s − 1.05·7-s + 4.38·9-s + 5.28·11-s − 6.46·13-s − 5.29·15-s + 2.74·17-s − 1.15·19-s + 2.86·21-s + 1.55·23-s − 1.20·25-s − 3.75·27-s + 1.46·29-s + 2.93·31-s − 14.3·33-s − 2.05·35-s − 9.73·37-s + 17.5·39-s − 10.8·41-s + 11.4·43-s + 8.53·45-s − 7.00·47-s − 5.88·49-s − 7.46·51-s + 1.16·53-s + 10.2·55-s + ⋯
L(s)  = 1  − 1.56·3-s + 0.871·5-s − 0.398·7-s + 1.46·9-s + 1.59·11-s − 1.79·13-s − 1.36·15-s + 0.666·17-s − 0.263·19-s + 0.624·21-s + 0.324·23-s − 0.240·25-s − 0.721·27-s + 0.271·29-s + 0.527·31-s − 2.49·33-s − 0.347·35-s − 1.60·37-s + 2.81·39-s − 1.68·41-s + 1.74·43-s + 1.27·45-s − 1.02·47-s − 0.841·49-s − 1.04·51-s + 0.160·53-s + 1.38·55-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 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 - 1.94T + 5T^{2} \)
7 \( 1 + 1.05T + 7T^{2} \)
11 \( 1 - 5.28T + 11T^{2} \)
13 \( 1 + 6.46T + 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 + 1.15T + 19T^{2} \)
23 \( 1 - 1.55T + 23T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 + 9.73T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 7.00T + 47T^{2} \)
53 \( 1 - 1.16T + 53T^{2} \)
59 \( 1 + 2.86T + 59T^{2} \)
61 \( 1 - 7.60T + 61T^{2} \)
67 \( 1 + 5.55T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 9.15T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 9.67T + 89T^{2} \)
97 \( 1 + 8.07T + 97T^{2} \)
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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

−7.35982482056572919155654693026, −6.72067807835675625756466545525, −6.38866068188481499978743285557, −5.55392671743049218522417719126, −5.06157047149546896429976407619, −4.32259193783099613402050101847, −3.28284634758021784138581254523, −2.07528950560215907189901407598, −1.18256119213195279486171195957, 0, 1.18256119213195279486171195957, 2.07528950560215907189901407598, 3.28284634758021784138581254523, 4.32259193783099613402050101847, 5.06157047149546896429976407619, 5.55392671743049218522417719126, 6.38866068188481499978743285557, 6.72067807835675625756466545525, 7.35982482056572919155654693026

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