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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.76·3-s + 0.187·5-s − 7-s + 0.124·9-s − 11-s − 13-s + 0.330·15-s + 4.32·17-s − 5.79·19-s − 1.76·21-s + 5.19·23-s − 4.96·25-s − 5.08·27-s + 1.25·29-s + 6.52·31-s − 1.76·33-s − 0.187·35-s − 5.96·37-s − 1.76·39-s + 5.23·41-s − 3.99·43-s + 0.0233·45-s − 5.21·47-s + 49-s + 7.64·51-s − 1.33·53-s − 0.187·55-s + ⋯
L(s)  = 1  + 1.02·3-s + 0.0837·5-s − 0.377·7-s + 0.0415·9-s − 0.301·11-s − 0.277·13-s + 0.0854·15-s + 1.04·17-s − 1.33·19-s − 0.385·21-s + 1.08·23-s − 0.992·25-s − 0.978·27-s + 0.233·29-s + 1.17·31-s − 0.307·33-s − 0.0316·35-s − 0.981·37-s − 0.283·39-s + 0.817·41-s − 0.609·43-s + 0.00347·45-s − 0.761·47-s + 0.142·49-s + 1.07·51-s − 0.183·53-s − 0.0252·55-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 \( 1 - 1.76T + 3T^{2} \)
5 \( 1 - 0.187T + 5T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
23 \( 1 - 5.19T + 23T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 - 6.52T + 31T^{2} \)
37 \( 1 + 5.96T + 37T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 + 3.99T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 + 1.33T + 53T^{2} \)
59 \( 1 - 7.28T + 59T^{2} \)
61 \( 1 + 3.40T + 61T^{2} \)
67 \( 1 + 2.43T + 67T^{2} \)
71 \( 1 + 2.65T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 3.58T + 79T^{2} \)
83 \( 1 + 9.04T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 4.31T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.60898071764560603237538443679, −6.91560788424068990734840695246, −6.12086010589713446218770441441, −5.43743811075816633216713027236, −4.55176307067779735222353947488, −3.73797860725752906589876368468, −2.99372050873294885968319230066, −2.45894774798311388434405949580, −1.44623589490272966392194170247, 0, 1.44623589490272966392194170247, 2.45894774798311388434405949580, 2.99372050873294885968319230066, 3.73797860725752906589876368468, 4.55176307067779735222353947488, 5.43743811075816633216713027236, 6.12086010589713446218770441441, 6.91560788424068990734840695246, 7.60898071764560603237538443679

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