Properties

Label 2-8008-1.1-c1-0-78
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  − 2.33·3-s − 2.59·5-s + 7-s + 2.43·9-s − 11-s + 13-s + 6.05·15-s + 0.771·17-s − 0.470·19-s − 2.33·21-s − 7.90·23-s + 1.74·25-s + 1.31·27-s − 3.36·29-s + 5.85·31-s + 2.33·33-s − 2.59·35-s + 7.25·37-s − 2.33·39-s − 10.7·41-s + 8.08·43-s − 6.33·45-s − 1.01·47-s + 49-s − 1.79·51-s − 2.08·53-s + 2.59·55-s + ⋯
L(s)  = 1  − 1.34·3-s − 1.16·5-s + 0.377·7-s + 0.812·9-s − 0.301·11-s + 0.277·13-s + 1.56·15-s + 0.187·17-s − 0.107·19-s − 0.508·21-s − 1.64·23-s + 0.348·25-s + 0.252·27-s − 0.624·29-s + 1.05·31-s + 0.405·33-s − 0.438·35-s + 1.19·37-s − 0.373·39-s − 1.67·41-s + 1.23·43-s − 0.943·45-s − 0.147·47-s + 0.142·49-s − 0.251·51-s − 0.286·53-s + 0.350·55-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 \( 1 + 2.33T + 3T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
17 \( 1 - 0.771T + 17T^{2} \)
19 \( 1 + 0.470T + 19T^{2} \)
23 \( 1 + 7.90T + 23T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 - 5.85T + 31T^{2} \)
37 \( 1 - 7.25T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 8.08T + 43T^{2} \)
47 \( 1 + 1.01T + 47T^{2} \)
53 \( 1 + 2.08T + 53T^{2} \)
59 \( 1 + 2.38T + 59T^{2} \)
61 \( 1 - 4.66T + 61T^{2} \)
67 \( 1 - 6.10T + 67T^{2} \)
71 \( 1 - 6.10T + 71T^{2} \)
73 \( 1 - 8.85T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 1.15T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−7.50077769166681426354131988026, −6.72527705021747755295081637341, −6.01529809220920638546445770559, −5.48767497017707976112621464756, −4.61024543471019041198142635087, −4.15836208944935024078773599216, −3.29883671168150239511851634183, −2.10094588999219675823987425008, −0.883450518125937748891886359227, 0, 0.883450518125937748891886359227, 2.10094588999219675823987425008, 3.29883671168150239511851634183, 4.15836208944935024078773599216, 4.61024543471019041198142635087, 5.48767497017707976112621464756, 6.01529809220920638546445770559, 6.72527705021747755295081637341, 7.50077769166681426354131988026

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