Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·3-s + 4.10·5-s − 7-s + 4.49·9-s + 11-s − 13-s − 11.2·15-s − 4.75·17-s − 6.50·19-s + 2.73·21-s − 1.11·23-s + 11.8·25-s − 4.09·27-s + 5.69·29-s + 9.59·31-s − 2.73·33-s − 4.10·35-s + 0.925·37-s + 2.73·39-s − 5.67·41-s − 5.94·43-s + 18.4·45-s − 1.11·47-s + 49-s + 13.0·51-s − 10.7·53-s + 4.10·55-s + ⋯
L(s)  = 1  − 1.58·3-s + 1.83·5-s − 0.377·7-s + 1.49·9-s + 0.301·11-s − 0.277·13-s − 2.90·15-s − 1.15·17-s − 1.49·19-s + 0.597·21-s − 0.233·23-s + 2.37·25-s − 0.787·27-s + 1.05·29-s + 1.72·31-s − 0.476·33-s − 0.693·35-s + 0.152·37-s + 0.438·39-s − 0.886·41-s − 0.907·43-s + 2.75·45-s − 0.163·47-s + 0.142·49-s + 1.82·51-s − 1.48·53-s + 0.553·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + 2.73T + 3T^{2} \)
5 \( 1 - 4.10T + 5T^{2} \)
17 \( 1 + 4.75T + 17T^{2} \)
19 \( 1 + 6.50T + 19T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 - 5.69T + 29T^{2} \)
31 \( 1 - 9.59T + 31T^{2} \)
37 \( 1 - 0.925T + 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 + 5.94T + 43T^{2} \)
47 \( 1 + 1.11T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 7.08T + 59T^{2} \)
61 \( 1 - 7.14T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 7.59T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 0.448T + 83T^{2} \)
89 \( 1 + 7.30T + 89T^{2} \)
97 \( 1 + 4.67T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.89447620700417861315637818941, −6.50115192301842927422554350252, −6.27238012657627418061571982841, −5.53927996948039130983074904186, −4.78558218403527930847206060646, −4.36725770430404540243768104343, −2.85972697727490256506850838814, −2.06554818872864209692774015771, −1.21182295029578173048264851115, 0, 1.21182295029578173048264851115, 2.06554818872864209692774015771, 2.85972697727490256506850838814, 4.36725770430404540243768104343, 4.78558218403527930847206060646, 5.53927996948039130983074904186, 6.27238012657627418061571982841, 6.50115192301842927422554350252, 6.89447620700417861315637818941

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