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  + 4·7-s − 4·11-s + 6·13-s − 2·17-s − 4·23-s − 5·25-s − 10·37-s − 6·41-s − 4·43-s + 6·47-s + 9·49-s + 6·53-s − 4·59-s − 12·61-s − 4·67-s − 6·71-s − 14·73-s − 16·77-s − 8·79-s + 4·83-s + 6·89-s + 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.834·23-s − 25-s − 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s − 1.53·61-s − 0.488·67-s − 0.712·71-s − 1.63·73-s − 1.82·77-s − 0.900·79-s + 0.439·83-s + 0.635·89-s + 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−17.01958094182719, −16.20871424346465, −15.67667728923920, −15.35790398617534, −14.67746023634055, −13.81109547993785, −13.65281002246826, −13.09767732525274, −12.03795550000705, −11.77379054449980, −10.99154653785541, −10.58043110415776, −10.12982463858369, −8.945206188485247, −8.599618590418927, −7.962895847071679, −7.551624422509277, −6.623438910282091, −5.770681522899481, −5.364930593789499, −4.522642893235182, −3.924081601870843, −3.004242835416763, −1.937976221742638, −1.457528446621464, 0, 1.457528446621464, 1.937976221742638, 3.004242835416763, 3.924081601870843, 4.522642893235182, 5.364930593789499, 5.770681522899481, 6.623438910282091, 7.551624422509277, 7.962895847071679, 8.599618590418927, 8.945206188485247, 10.12982463858369, 10.58043110415776, 10.99154653785541, 11.77379054449980, 12.03795550000705, 13.09767732525274, 13.65281002246826, 13.81109547993785, 14.67746023634055, 15.35790398617534, 15.67667728923920, 16.20871424346465, 17.01958094182719

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