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  + 2.18·3-s − 2.05·5-s + 7-s + 1.78·9-s − 11-s − 13-s − 4.50·15-s + 1.94·17-s + 2.63·19-s + 2.18·21-s − 2.37·23-s − 0.757·25-s − 2.65·27-s + 8.17·29-s + 0.915·31-s − 2.18·33-s − 2.05·35-s + 0.853·37-s − 2.18·39-s + 6.33·41-s − 8.24·43-s − 3.67·45-s − 4.20·47-s + 49-s + 4.24·51-s + 6.84·53-s + 2.05·55-s + ⋯
L(s)  = 1  + 1.26·3-s − 0.921·5-s + 0.377·7-s + 0.595·9-s − 0.301·11-s − 0.277·13-s − 1.16·15-s + 0.470·17-s + 0.604·19-s + 0.477·21-s − 0.495·23-s − 0.151·25-s − 0.511·27-s + 1.51·29-s + 0.164·31-s − 0.380·33-s − 0.348·35-s + 0.140·37-s − 0.350·39-s + 0.989·41-s − 1.25·43-s − 0.548·45-s − 0.613·47-s + 0.142·49-s + 0.594·51-s + 0.940·53-s + 0.277·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.671550881$
$L(\frac12)$  $\approx$  $2.671550881$
$L(\frac{3}{2})$   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 \( 1 - 2.18T + 3T^{2} \)
5 \( 1 + 2.05T + 5T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 - 2.63T + 19T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 - 0.915T + 31T^{2} \)
37 \( 1 - 0.853T + 37T^{2} \)
41 \( 1 - 6.33T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + 4.20T + 47T^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 2.58T + 71T^{2} \)
73 \( 1 + 9.32T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 3.62T + 83T^{2} \)
89 \( 1 + 7.74T + 89T^{2} \)
97 \( 1 - 4.78T + 97T^{2} \)
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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

−7.974261582306875417037006588466, −7.43609375514746182365880967305, −6.70046672858981551277919136008, −5.66099549780524423442750464620, −4.88243675539523137050735881551, −4.08344876876554845633130017447, −3.46971637601482508249585324428, −2.76450242933116034936924820803, −1.99599700706837008135856648721, −0.75236697924852741213140753057, 0.75236697924852741213140753057, 1.99599700706837008135856648721, 2.76450242933116034936924820803, 3.46971637601482508249585324428, 4.08344876876554845633130017447, 4.88243675539523137050735881551, 5.66099549780524423442750464620, 6.70046672858981551277919136008, 7.43609375514746182365880967305, 7.974261582306875417037006588466

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