Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08·3-s − 0.239·5-s + 7-s + 6.51·9-s − 11-s − 13-s + 0.738·15-s + 3.76·17-s − 6.06·19-s − 3.08·21-s − 0.585·23-s − 4.94·25-s − 10.8·27-s − 8.13·29-s + 10.9·31-s + 3.08·33-s − 0.239·35-s + 3.95·37-s + 3.08·39-s + 12.3·41-s − 9.90·43-s − 1.55·45-s − 4.96·47-s + 49-s − 11.5·51-s + 9.75·53-s + 0.239·55-s + ⋯
L(s)  = 1  − 1.78·3-s − 0.107·5-s + 0.377·7-s + 2.17·9-s − 0.301·11-s − 0.277·13-s + 0.190·15-s + 0.912·17-s − 1.39·19-s − 0.673·21-s − 0.122·23-s − 0.988·25-s − 2.08·27-s − 1.50·29-s + 1.96·31-s + 0.536·33-s − 0.0404·35-s + 0.649·37-s + 0.493·39-s + 1.93·41-s − 1.51·43-s − 0.232·45-s − 0.724·47-s + 0.142·49-s − 1.62·51-s + 1.33·53-s + 0.0322·55-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  =  0
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7308779876$
$L(\frac12)$  $\approx$  $0.7308779876$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 + 0.239T + 5T^{2} \)
17 \( 1 - 3.76T + 17T^{2} \)
19 \( 1 + 6.06T + 19T^{2} \)
23 \( 1 + 0.585T + 23T^{2} \)
29 \( 1 + 8.13T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 - 3.95T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 9.90T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 - 9.75T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 - 9.25T + 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 - 8.56T + 79T^{2} \)
83 \( 1 - 2.51T + 83T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
97 \( 1 - 0.759T + 97T^{2} \)
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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

−7.83168007380794387022751248684, −6.93782234144864317773957149391, −6.34200266664885988464503641334, −5.71379660659412055140918269091, −5.20180515668279148869463077093, −4.40901474017149812705305060317, −3.89687989572956842627452935644, −2.50508175826909749611089912421, −1.50276820321548057916392226687, −0.48455994677648237936127141533, 0.48455994677648237936127141533, 1.50276820321548057916392226687, 2.50508175826909749611089912421, 3.89687989572956842627452935644, 4.40901474017149812705305060317, 5.20180515668279148869463077093, 5.71379660659412055140918269091, 6.34200266664885988464503641334, 6.93782234144864317773957149391, 7.83168007380794387022751248684

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