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 − 5·7-s + 11-s + 5·13-s + 2·17-s + 2·19-s − 4·23-s − 4·25-s − 7·29-s − 7·31-s + 5·35-s + 6·37-s + 6·41-s + 6·43-s + 8·47-s + 18·49-s + 4·53-s − 55-s − 6·59-s − 6·61-s − 5·65-s + 5·67-s + 71-s − 4·73-s − 5·77-s − 17·79-s + 15·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.88·7-s + 0.301·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s − 4/5·25-s − 1.29·29-s − 1.25·31-s + 0.845·35-s + 0.986·37-s + 0.937·41-s + 0.914·43-s + 1.16·47-s + 18/7·49-s + 0.549·53-s − 0.134·55-s − 0.781·59-s − 0.768·61-s − 0.620·65-s + 0.610·67-s + 0.118·71-s − 0.468·73-s − 0.569·77-s − 1.91·79-s + 1.64·83-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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.71453592170783, −16.21704940202432, −15.90712212697922, −15.44985344872844, −14.63580124462090, −13.99136884442272, −13.27776894766378, −13.04387741621824, −12.28160253429498, −11.83007220945905, −11.01412421230322, −10.54986510984568, −9.697581324764620, −9.298970184380495, −8.826554083229556, −7.734452254854643, −7.444346918131601, −6.532198583572608, −5.862737660776025, −5.704707607191429, −4.149322723034036, −3.774823334415856, −3.250341354992418, −2.263991171816911, −1.036630694613953, 0, 1.036630694613953, 2.263991171816911, 3.250341354992418, 3.774823334415856, 4.149322723034036, 5.704707607191429, 5.862737660776025, 6.532198583572608, 7.444346918131601, 7.734452254854643, 8.826554083229556, 9.298970184380495, 9.697581324764620, 10.54986510984568, 11.01412421230322, 11.83007220945905, 12.28160253429498, 13.04387741621824, 13.27776894766378, 13.99136884442272, 14.63580124462090, 15.44985344872844, 15.90712212697922, 16.21704940202432, 16.71453592170783

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