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  − 4·5-s − 2·7-s − 5·11-s − 13-s + 5·17-s − 7·19-s + 8·23-s + 11·25-s + 8·29-s − 10·31-s + 8·35-s + 3·37-s + 12·41-s − 3·43-s − 47-s − 3·49-s − 5·53-s + 20·55-s + 12·61-s + 4·65-s + 2·67-s + 4·71-s + 2·73-s + 10·77-s − 14·79-s − 20·85-s + 12·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.755·7-s − 1.50·11-s − 0.277·13-s + 1.21·17-s − 1.60·19-s + 1.66·23-s + 11/5·25-s + 1.48·29-s − 1.79·31-s + 1.35·35-s + 0.493·37-s + 1.87·41-s − 0.457·43-s − 0.145·47-s − 3/7·49-s − 0.686·53-s + 2.69·55-s + 1.53·61-s + 0.496·65-s + 0.244·67-s + 0.474·71-s + 0.234·73-s + 1.13·77-s − 1.57·79-s − 2.16·85-s + 1.27·89-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
139 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−16.63022708436632, −16.30377709644534, −15.79602652472497, −15.31794303395507, −14.69399354222121, −14.36536742532462, −13.05244938148452, −12.77851989502109, −12.56354995835073, −11.67051674420809, −11.01307048861921, −10.66743381821267, −9.981399182435190, −9.144181842619312, −8.418398113475041, −7.970085685352882, −7.357458907072515, −6.881741425345772, −5.977239310580060, −5.076801667026641, −4.566127523642891, −3.691349256406560, −3.115792249997701, −2.465006857660561, −0.8305340453178465, 0, 0.8305340453178465, 2.465006857660561, 3.115792249997701, 3.691349256406560, 4.566127523642891, 5.076801667026641, 5.977239310580060, 6.881741425345772, 7.357458907072515, 7.970085685352882, 8.418398113475041, 9.144181842619312, 9.981399182435190, 10.66743381821267, 11.01307048861921, 11.67051674420809, 12.56354995835073, 12.77851989502109, 13.05244938148452, 14.36536742532462, 14.69399354222121, 15.31794303395507, 15.79602652472497, 16.30377709644534, 16.63022708436632

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