Properties

Label 2-8008-1.1-c1-0-116
Degree $2$
Conductor $8008$
Sign $-1$
Analytic cond. $63.9442$
Root an. cond. $7.99651$
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  − 2.59·3-s + 0.486·5-s + 7-s + 3.73·9-s + 11-s + 13-s − 1.26·15-s + 4.76·17-s − 5.64·19-s − 2.59·21-s − 5.89·23-s − 4.76·25-s − 1.90·27-s + 5.02·29-s + 7.10·31-s − 2.59·33-s + 0.486·35-s + 3.80·37-s − 2.59·39-s − 5.77·41-s − 1.96·43-s + 1.81·45-s − 3.82·47-s + 49-s − 12.3·51-s + 1.43·53-s + 0.486·55-s + ⋯
L(s)  = 1  − 1.49·3-s + 0.217·5-s + 0.377·7-s + 1.24·9-s + 0.301·11-s + 0.277·13-s − 0.326·15-s + 1.15·17-s − 1.29·19-s − 0.566·21-s − 1.22·23-s − 0.952·25-s − 0.367·27-s + 0.932·29-s + 1.27·31-s − 0.451·33-s + 0.0823·35-s + 0.626·37-s − 0.415·39-s − 0.902·41-s − 0.299·43-s + 0.271·45-s − 0.558·47-s + 0.142·49-s − 1.73·51-s + 0.197·53-s + 0.0656·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(63.9442\)
Root analytic conductor: \(7.99651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8008,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + 2.59T + 3T^{2} \)
5 \( 1 - 0.486T + 5T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 + 5.89T + 23T^{2} \)
29 \( 1 - 5.02T + 29T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 - 3.80T + 37T^{2} \)
41 \( 1 + 5.77T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 - 1.43T + 53T^{2} \)
59 \( 1 + 5.08T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 7.40T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 4.42T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 9.76T + 97T^{2} \)
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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

−7.41968509530975529253432367134, −6.38726199086536077407880714444, −6.18867599471717976756719255715, −5.56146156175283862299439203611, −4.66243915618972031239014961011, −4.26331715169386929041235931466, −3.16811059016404385167860921953, −1.95186565136203796041376299459, −1.12396990776620900929751655106, 0, 1.12396990776620900929751655106, 1.95186565136203796041376299459, 3.16811059016404385167860921953, 4.26331715169386929041235931466, 4.66243915618972031239014961011, 5.56146156175283862299439203611, 6.18867599471717976756719255715, 6.38726199086536077407880714444, 7.41968509530975529253432367134

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