Properties

Label 2-6008-1.1-c1-0-150
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  + 3.00·3-s − 3.81·5-s − 4.40·7-s + 6.05·9-s + 1.92·11-s + 5.13·13-s − 11.4·15-s − 0.577·17-s + 0.576·19-s − 13.2·21-s − 2.13·23-s + 9.51·25-s + 9.19·27-s − 2.65·29-s − 8.39·31-s + 5.79·33-s + 16.8·35-s − 4.69·37-s + 15.4·39-s − 11.4·41-s + 9.20·43-s − 23.0·45-s + 2.10·47-s + 12.4·49-s − 1.73·51-s + 13.3·53-s − 7.33·55-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.70·5-s − 1.66·7-s + 2.01·9-s + 0.580·11-s + 1.42·13-s − 2.96·15-s − 0.140·17-s + 0.132·19-s − 2.89·21-s − 0.444·23-s + 1.90·25-s + 1.76·27-s − 0.493·29-s − 1.50·31-s + 1.00·33-s + 2.84·35-s − 0.771·37-s + 2.47·39-s − 1.78·41-s + 1.40·43-s − 3.43·45-s + 0.307·47-s + 1.77·49-s − 0.243·51-s + 1.83·53-s − 0.989·55-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 \( 1 - 3.00T + 3T^{2} \)
5 \( 1 + 3.81T + 5T^{2} \)
7 \( 1 + 4.40T + 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 - 5.13T + 13T^{2} \)
17 \( 1 + 0.577T + 17T^{2} \)
19 \( 1 - 0.576T + 19T^{2} \)
23 \( 1 + 2.13T + 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 + 8.39T + 31T^{2} \)
37 \( 1 + 4.69T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 9.20T + 43T^{2} \)
47 \( 1 - 2.10T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 4.21T + 59T^{2} \)
61 \( 1 + 2.63T + 61T^{2} \)
67 \( 1 + 5.96T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−7.82090723979639046298390912271, −7.11365624904634366293974051593, −6.73200901107404210751464771615, −5.67203446332798383712352647106, −4.14238629018387692686869022332, −3.79985562498066487452987158131, −3.44145561256838752345817138710, −2.70987904665280436306941441130, −1.41160987602381535266572690238, 0, 1.41160987602381535266572690238, 2.70987904665280436306941441130, 3.44145561256838752345817138710, 3.79985562498066487452987158131, 4.14238629018387692686869022332, 5.67203446332798383712352647106, 6.73200901107404210751464771615, 7.11365624904634366293974051593, 7.82090723979639046298390912271

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