Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 139 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 3·11-s − 5·13-s + 8·23-s − 4·25-s − 3·29-s + 3·31-s − 35-s − 10·37-s + 2·41-s + 10·43-s − 4·47-s − 6·49-s − 6·53-s − 3·55-s − 6·59-s − 6·61-s + 5·65-s + 5·67-s − 3·71-s − 2·73-s + 3·77-s − 3·79-s + 17·83-s + 9·89-s − 5·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.904·11-s − 1.38·13-s + 1.66·23-s − 4/5·25-s − 0.557·29-s + 0.538·31-s − 0.169·35-s − 1.64·37-s + 0.312·41-s + 1.52·43-s − 0.583·47-s − 6/7·49-s − 0.824·53-s − 0.404·55-s − 0.781·59-s − 0.768·61-s + 0.620·65-s + 0.610·67-s − 0.356·71-s − 0.234·73-s + 0.341·77-s − 0.337·79-s + 1.86·83-s + 0.953·89-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  =  1
Selberg data  =  $(2,\ 10008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;139\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.98590427000071, −16.45817357854213, −15.63375579033415, −15.23473900488396, −14.55872065044519, −14.27455716919986, −13.50728389175900, −12.78403220434020, −12.17208608738889, −11.82562376298521, −11.07910578491263, −10.65090462667448, −9.681353296682247, −9.321758739824451, −8.660019576581538, −7.825448870403683, −7.372413500863761, −6.761016940695273, −6.000795444130883, −5.040821724506133, −4.680951576504720, −3.794185831964092, −3.073428865401004, −2.138727075477678, −1.221141727208652, 0, 1.221141727208652, 2.138727075477678, 3.073428865401004, 3.794185831964092, 4.680951576504720, 5.040821724506133, 6.000795444130883, 6.761016940695273, 7.372413500863761, 7.825448870403683, 8.660019576581538, 9.321758739824451, 9.681353296682247, 10.65090462667448, 11.07910578491263, 11.82562376298521, 12.17208608738889, 12.78403220434020, 13.50728389175900, 14.27455716919986, 14.55872065044519, 15.23473900488396, 15.63375579033415, 16.45817357854213, 16.98590427000071

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