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  + 2.82·3-s + 0.233·5-s − 7-s + 4.99·9-s + 11-s − 13-s + 0.659·15-s − 1.37·17-s − 2.67·19-s − 2.82·21-s − 3.07·23-s − 4.94·25-s + 5.64·27-s + 9.22·29-s + 6.01·31-s + 2.82·33-s − 0.233·35-s − 4.26·37-s − 2.82·39-s + 8.07·41-s + 6.43·43-s + 1.16·45-s + 11.0·47-s + 49-s − 3.89·51-s + 3.87·53-s + 0.233·55-s + ⋯
L(s)  = 1  + 1.63·3-s + 0.104·5-s − 0.377·7-s + 1.66·9-s + 0.301·11-s − 0.277·13-s + 0.170·15-s − 0.333·17-s − 0.614·19-s − 0.617·21-s − 0.641·23-s − 0.989·25-s + 1.08·27-s + 1.71·29-s + 1.08·31-s + 0.492·33-s − 0.0394·35-s − 0.700·37-s − 0.452·39-s + 1.26·41-s + 0.981·43-s + 0.173·45-s + 1.61·47-s + 0.142·49-s − 0.545·51-s + 0.532·53-s + 0.0314·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.851951616$
$L(\frac12)$  $\approx$  $3.851951616$
$L(\frac{3}{2})$   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 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 - 0.233T + 5T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 + 2.67T + 19T^{2} \)
23 \( 1 + 3.07T + 23T^{2} \)
29 \( 1 - 9.22T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 + 4.26T + 37T^{2} \)
41 \( 1 - 8.07T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 3.87T + 53T^{2} \)
59 \( 1 + 9.61T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 0.633T + 71T^{2} \)
73 \( 1 - 6.61T + 73T^{2} \)
79 \( 1 - 1.86T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−7.958862784506032282838922084672, −7.32979104666789545444956490673, −6.54830370734084552187227191202, −5.94314580396471500102335728627, −4.75331540603470013021221139983, −4.06856131152809673171384879342, −3.50931694238581275965121154683, −2.44510121976464074044270484972, −2.25989157905654449193559247134, −0.896102107191924041388679186407, 0.896102107191924041388679186407, 2.25989157905654449193559247134, 2.44510121976464074044270484972, 3.50931694238581275965121154683, 4.06856131152809673171384879342, 4.75331540603470013021221139983, 5.94314580396471500102335728627, 6.54830370734084552187227191202, 7.32979104666789545444956490673, 7.958862784506032282838922084672

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