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  − 1.97·3-s − 1.63·5-s + 7-s + 0.895·9-s − 11-s − 13-s + 3.22·15-s + 7.91·17-s − 4.55·19-s − 1.97·21-s − 1.32·23-s − 2.33·25-s + 4.15·27-s + 5.69·29-s − 4.04·31-s + 1.97·33-s − 1.63·35-s − 10.2·37-s + 1.97·39-s + 6.71·41-s − 1.42·43-s − 1.46·45-s − 8.30·47-s + 49-s − 15.6·51-s + 1.91·53-s + 1.63·55-s + ⋯
L(s)  = 1  − 1.13·3-s − 0.729·5-s + 0.377·7-s + 0.298·9-s − 0.301·11-s − 0.277·13-s + 0.831·15-s + 1.91·17-s − 1.04·19-s − 0.430·21-s − 0.275·23-s − 0.467·25-s + 0.799·27-s + 1.05·29-s − 0.727·31-s + 0.343·33-s − 0.275·35-s − 1.68·37-s + 0.316·39-s + 1.04·41-s − 0.217·43-s − 0.217·45-s − 1.21·47-s + 0.142·49-s − 2.18·51-s + 0.262·53-s + 0.220·55-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 \( 1 + 1.97T + 3T^{2} \)
5 \( 1 + 1.63T + 5T^{2} \)
17 \( 1 - 7.91T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 + 1.32T + 23T^{2} \)
29 \( 1 - 5.69T + 29T^{2} \)
31 \( 1 + 4.04T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 6.71T + 41T^{2} \)
43 \( 1 + 1.42T + 43T^{2} \)
47 \( 1 + 8.30T + 47T^{2} \)
53 \( 1 - 1.91T + 53T^{2} \)
59 \( 1 - 3.45T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 5.48T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 7.86T + 89T^{2} \)
97 \( 1 + 5.14T + 97T^{2} \)
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

−7.48011459477042243764990817502, −6.75104151927799582082408708711, −6.02654107949429658254501560346, −5.32643926661604397254594837721, −4.90107009508469371664180990532, −3.96259516257739113471199249057, −3.27508700998887509559485956896, −2.12810362270126407858937287247, −0.968692232158686932932639130420, 0, 0.968692232158686932932639130420, 2.12810362270126407858937287247, 3.27508700998887509559485956896, 3.96259516257739113471199249057, 4.90107009508469371664180990532, 5.32643926661604397254594837721, 6.02654107949429658254501560346, 6.75104151927799582082408708711, 7.48011459477042243764990817502

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