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  − 2.37·3-s − 0.155·5-s − 7-s + 2.64·9-s + 11-s + 13-s + 0.369·15-s − 3.88·17-s − 3.56·19-s + 2.37·21-s + 5.09·23-s − 4.97·25-s + 0.849·27-s − 6.55·29-s + 4.73·31-s − 2.37·33-s + 0.155·35-s + 7.33·37-s − 2.37·39-s + 11.7·41-s + 7.29·43-s − 0.411·45-s − 9.56·47-s + 49-s + 9.23·51-s − 8.39·53-s − 0.155·55-s + ⋯
L(s)  = 1  − 1.37·3-s − 0.0696·5-s − 0.377·7-s + 0.880·9-s + 0.301·11-s + 0.277·13-s + 0.0954·15-s − 0.943·17-s − 0.818·19-s + 0.518·21-s + 1.06·23-s − 0.995·25-s + 0.163·27-s − 1.21·29-s + 0.850·31-s − 0.413·33-s + 0.0263·35-s + 1.20·37-s − 0.380·39-s + 1.84·41-s + 1.11·43-s − 0.0613·45-s − 1.39·47-s + 0.142·49-s + 1.29·51-s − 1.15·53-s − 0.0209·55-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 \( 1 + 2.37T + 3T^{2} \)
5 \( 1 + 0.155T + 5T^{2} \)
17 \( 1 + 3.88T + 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 + 6.55T + 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 - 7.33T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 - 7.29T + 43T^{2} \)
47 \( 1 + 9.56T + 47T^{2} \)
53 \( 1 + 8.39T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 8.66T + 61T^{2} \)
67 \( 1 + 9.49T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 + 3.14T + 79T^{2} \)
83 \( 1 - 7.49T + 83T^{2} \)
89 \( 1 - 0.972T + 89T^{2} \)
97 \( 1 + 4.23T + 97T^{2} \)
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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

−7.32322072957569979169435867200, −6.56677555745726382600595497198, −6.09507726461805120432265097810, −5.62583138175127792569438768058, −4.55019991270548767022584369218, −4.27232550779958299645048135170, −3.13710020209251741921458513622, −2.12618314041601491093148987829, −0.971064677565740629932276614424, 0, 0.971064677565740629932276614424, 2.12618314041601491093148987829, 3.13710020209251741921458513622, 4.27232550779958299645048135170, 4.55019991270548767022584369218, 5.62583138175127792569438768058, 6.09507726461805120432265097810, 6.56677555745726382600595497198, 7.32322072957569979169435867200

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