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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.847·3-s − 2.05·5-s + 7-s − 2.28·9-s − 11-s − 13-s − 1.74·15-s + 7.00·17-s − 7.01·19-s + 0.847·21-s − 0.954·23-s − 0.762·25-s − 4.47·27-s + 1.08·29-s + 1.96·31-s − 0.847·33-s − 2.05·35-s + 9.98·37-s − 0.847·39-s + 0.778·41-s + 5.80·43-s + 4.69·45-s − 0.446·47-s + 49-s + 5.93·51-s − 9.47·53-s + 2.05·55-s + ⋯
L(s)  = 1  + 0.489·3-s − 0.920·5-s + 0.377·7-s − 0.760·9-s − 0.301·11-s − 0.277·13-s − 0.450·15-s + 1.69·17-s − 1.60·19-s + 0.184·21-s − 0.199·23-s − 0.152·25-s − 0.861·27-s + 0.201·29-s + 0.352·31-s − 0.147·33-s − 0.347·35-s + 1.64·37-s − 0.135·39-s + 0.121·41-s + 0.884·43-s + 0.700·45-s − 0.0650·47-s + 0.142·49-s + 0.830·51-s − 1.30·53-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.473153563$
$L(\frac12)$  $\approx$  $1.473153563$
$L(\frac{3}{2})$   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 \( 1 - 0.847T + 3T^{2} \)
5 \( 1 + 2.05T + 5T^{2} \)
17 \( 1 - 7.00T + 17T^{2} \)
19 \( 1 + 7.01T + 19T^{2} \)
23 \( 1 + 0.954T + 23T^{2} \)
29 \( 1 - 1.08T + 29T^{2} \)
31 \( 1 - 1.96T + 31T^{2} \)
37 \( 1 - 9.98T + 37T^{2} \)
41 \( 1 - 0.778T + 41T^{2} \)
43 \( 1 - 5.80T + 43T^{2} \)
47 \( 1 + 0.446T + 47T^{2} \)
53 \( 1 + 9.47T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 + 2.84T + 67T^{2} \)
71 \( 1 + 3.58T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 2.31T + 97T^{2} \)
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

−7.924140386352757531991833243355, −7.56232612618368060777345551610, −6.32677459463757091324687641395, −5.87409323548935088394512958752, −4.87209077004891240881278621675, −4.26712419429564901664945242750, −3.43337663382470609415328164394, −2.80180844039832847671645037320, −1.89009025552844487052865830224, −0.56862514794612938016278200461, 0.56862514794612938016278200461, 1.89009025552844487052865830224, 2.80180844039832847671645037320, 3.43337663382470609415328164394, 4.26712419429564901664945242750, 4.87209077004891240881278621675, 5.87409323548935088394512958752, 6.32677459463757091324687641395, 7.56232612618368060777345551610, 7.924140386352757531991833243355

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