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  − 2.98·3-s + 3.19·5-s + 7-s + 5.89·9-s − 11-s − 13-s − 9.53·15-s + 1.18·17-s + 0.300·19-s − 2.98·21-s − 7.32·23-s + 5.22·25-s − 8.64·27-s + 3.24·29-s − 6.56·31-s + 2.98·33-s + 3.19·35-s − 4.73·37-s + 2.98·39-s − 4.72·41-s + 8.34·43-s + 18.8·45-s − 3.02·47-s + 49-s − 3.54·51-s + 9.50·53-s − 3.19·55-s + ⋯
L(s)  = 1  − 1.72·3-s + 1.42·5-s + 0.377·7-s + 1.96·9-s − 0.301·11-s − 0.277·13-s − 2.46·15-s + 0.287·17-s + 0.0689·19-s − 0.650·21-s − 1.52·23-s + 1.04·25-s − 1.66·27-s + 0.601·29-s − 1.17·31-s + 0.519·33-s + 0.540·35-s − 0.779·37-s + 0.477·39-s − 0.737·41-s + 1.27·43-s + 2.81·45-s − 0.441·47-s + 0.142·49-s − 0.495·51-s + 1.30·53-s − 0.431·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.328087680$
$L(\frac12)$  $\approx$  $1.328087680$
$L(\frac{3}{2})$   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 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 - 3.19T + 5T^{2} \)
17 \( 1 - 1.18T + 17T^{2} \)
19 \( 1 - 0.300T + 19T^{2} \)
23 \( 1 + 7.32T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 + 4.73T + 37T^{2} \)
41 \( 1 + 4.72T + 41T^{2} \)
43 \( 1 - 8.34T + 43T^{2} \)
47 \( 1 + 3.02T + 47T^{2} \)
53 \( 1 - 9.50T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 1.00T + 61T^{2} \)
67 \( 1 + 7.44T + 67T^{2} \)
71 \( 1 + 3.06T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 - 5.83T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.59443473964553056105233373656, −6.91627230615183361385835596575, −6.23239484319941394711903058098, −5.68717310835992034418653297168, −5.31016494862943172496046375068, −4.64634474782400290054159206076, −3.70611252701762111099767429535, −2.29846856472839631272994129202, −1.69096114458614412316609926817, −0.63332007259287476634021488357, 0.63332007259287476634021488357, 1.69096114458614412316609926817, 2.29846856472839631272994129202, 3.70611252701762111099767429535, 4.64634474782400290054159206076, 5.31016494862943172496046375068, 5.68717310835992034418653297168, 6.23239484319941394711903058098, 6.91627230615183361385835596575, 7.59443473964553056105233373656

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