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  − 3.38·3-s − 4.16·5-s + 7-s + 8.47·9-s − 11-s − 13-s + 14.1·15-s − 4.56·17-s − 0.578·19-s − 3.38·21-s − 6.33·23-s + 12.3·25-s − 18.5·27-s + 2.97·29-s + 5.14·31-s + 3.38·33-s − 4.16·35-s + 3.64·37-s + 3.38·39-s − 8.27·41-s − 10.1·43-s − 35.3·45-s − 3.73·47-s + 49-s + 15.4·51-s − 7.91·53-s + 4.16·55-s + ⋯
L(s)  = 1  − 1.95·3-s − 1.86·5-s + 0.377·7-s + 2.82·9-s − 0.301·11-s − 0.277·13-s + 3.64·15-s − 1.10·17-s − 0.132·19-s − 0.739·21-s − 1.32·23-s + 2.46·25-s − 3.57·27-s + 0.551·29-s + 0.924·31-s + 0.589·33-s − 0.703·35-s + 0.599·37-s + 0.542·39-s − 1.29·41-s − 1.55·43-s − 5.26·45-s − 0.545·47-s + 0.142·49-s + 2.16·51-s − 1.08·53-s + 0.561·55-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 \( 1 + 3.38T + 3T^{2} \)
5 \( 1 + 4.16T + 5T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 + 0.578T + 19T^{2} \)
23 \( 1 + 6.33T + 23T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 - 5.14T + 31T^{2} \)
37 \( 1 - 3.64T + 37T^{2} \)
41 \( 1 + 8.27T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 3.73T + 47T^{2} \)
53 \( 1 + 7.91T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 + 2.44T + 61T^{2} \)
67 \( 1 + 1.18T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 - 9.93T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.34979902046349258140028016234, −6.69877519881971587199602801162, −6.28028574504626011841372518774, −5.16649400424276325776230718502, −4.70803283823527116973990895040, −4.24386637902230813842579806039, −3.44794288695662002401175431837, −1.97372812087701460454186639777, −0.72222871217921031744753727822, 0, 0.72222871217921031744753727822, 1.97372812087701460454186639777, 3.44794288695662002401175431837, 4.24386637902230813842579806039, 4.70803283823527116973990895040, 5.16649400424276325776230718502, 6.28028574504626011841372518774, 6.69877519881971587199602801162, 7.34979902046349258140028016234

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