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  + 2.67·3-s − 4.01·5-s − 7-s + 4.13·9-s + 11-s + 13-s − 10.7·15-s − 6.26·17-s + 7.26·19-s − 2.67·21-s − 3.24·23-s + 11.1·25-s + 3.04·27-s + 0.715·29-s + 2.26·31-s + 2.67·33-s + 4.01·35-s − 5.56·37-s + 2.67·39-s + 6.82·41-s − 0.466·43-s − 16.6·45-s − 2.75·47-s + 49-s − 16.7·51-s − 11.5·53-s − 4.01·55-s + ⋯
L(s)  = 1  + 1.54·3-s − 1.79·5-s − 0.377·7-s + 1.37·9-s + 0.301·11-s + 0.277·13-s − 2.77·15-s − 1.51·17-s + 1.66·19-s − 0.583·21-s − 0.676·23-s + 2.22·25-s + 0.585·27-s + 0.132·29-s + 0.406·31-s + 0.465·33-s + 0.678·35-s − 0.914·37-s + 0.427·39-s + 1.06·41-s − 0.0711·43-s − 2.47·45-s − 0.401·47-s + 0.142·49-s − 2.34·51-s − 1.57·53-s − 0.541·55-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 \( 1 - 2.67T + 3T^{2} \)
5 \( 1 + 4.01T + 5T^{2} \)
17 \( 1 + 6.26T + 17T^{2} \)
19 \( 1 - 7.26T + 19T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 - 0.715T + 29T^{2} \)
31 \( 1 - 2.26T + 31T^{2} \)
37 \( 1 + 5.56T + 37T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 + 0.466T + 43T^{2} \)
47 \( 1 + 2.75T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 3.04T + 61T^{2} \)
67 \( 1 - 8.56T + 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 + 0.308T + 73T^{2} \)
79 \( 1 - 9.30T + 79T^{2} \)
83 \( 1 + 7.19T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 8.50T + 97T^{2} \)
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

−7.67796955850884526675378334169, −7.05936606038960911293970090268, −6.44907902882107295788646648025, −5.14284508681117693612314778331, −4.23296874613532818103816110597, −3.84327202311309227617272668437, −3.17357111146747014823982874503, −2.56268395819645378475836752948, −1.32947067015154340542839994478, 0, 1.32947067015154340542839994478, 2.56268395819645378475836752948, 3.17357111146747014823982874503, 3.84327202311309227617272668437, 4.23296874613532818103816110597, 5.14284508681117693612314778331, 6.44907902882107295788646648025, 7.05936606038960911293970090268, 7.67796955850884526675378334169

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