Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 139 $
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  + 3·5-s + 5·7-s − 5·11-s − 13-s + 8·19-s + 4·25-s + 29-s + 7·31-s + 15·35-s − 2·37-s + 10·41-s − 6·43-s − 12·47-s + 18·49-s + 6·53-s − 15·55-s + 6·59-s − 6·61-s − 3·65-s + 5·67-s − 7·71-s − 14·73-s − 25·77-s + 79-s + 83-s − 15·89-s − 5·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.88·7-s − 1.50·11-s − 0.277·13-s + 1.83·19-s + 4/5·25-s + 0.185·29-s + 1.25·31-s + 2.53·35-s − 0.328·37-s + 1.56·41-s − 0.914·43-s − 1.75·47-s + 18/7·49-s + 0.824·53-s − 2.02·55-s + 0.781·59-s − 0.768·61-s − 0.372·65-s + 0.610·67-s − 0.830·71-s − 1.63·73-s − 2.84·77-s + 0.112·79-s + 0.109·83-s − 1.58·89-s − 0.524·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10008\)    =    \(2^{3} \cdot 3^{2} \cdot 139\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10008} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.642239319$
$L(\frac12)$  $\approx$  $3.642239319$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;139\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;139\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
139 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−16.75769096028693, −16.01613197172772, −15.51537106534633, −14.73305420582626, −14.32866957151109, −13.71644812715756, −13.43437949593046, −12.69772342025317, −11.82563646460085, −11.43290994071780, −10.75534722700564, −9.974112218095449, −9.903524384971541, −8.816893275300600, −8.290343630067133, −7.623325227627101, −7.211949678494702, −6.024413599866504, −5.533810283563518, −4.947784158120465, −4.600205658841172, −3.160788723443505, −2.439703836265109, −1.767840836487812, −0.9575115747956255, 0.9575115747956255, 1.767840836487812, 2.439703836265109, 3.160788723443505, 4.600205658841172, 4.947784158120465, 5.533810283563518, 6.024413599866504, 7.211949678494702, 7.623325227627101, 8.290343630067133, 8.816893275300600, 9.903524384971541, 9.974112218095449, 10.75534722700564, 11.43290994071780, 11.82563646460085, 12.69772342025317, 13.43437949593046, 13.71644812715756, 14.32866957151109, 14.73305420582626, 15.51537106534633, 16.01613197172772, 16.75769096028693

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