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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 3·7-s + 3·11-s − 3·13-s + 6·17-s + 2·19-s + 8·23-s + 4·25-s + 5·29-s − 31-s − 9·35-s − 10·37-s − 2·41-s + 6·43-s + 8·47-s + 2·49-s + 4·53-s + 9·55-s + 2·59-s − 2·61-s − 9·65-s + 11·67-s + 3·71-s − 8·73-s − 9·77-s − 15·79-s − 11·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.13·7-s + 0.904·11-s − 0.832·13-s + 1.45·17-s + 0.458·19-s + 1.66·23-s + 4/5·25-s + 0.928·29-s − 0.179·31-s − 1.52·35-s − 1.64·37-s − 0.312·41-s + 0.914·43-s + 1.16·47-s + 2/7·49-s + 0.549·53-s + 1.21·55-s + 0.260·59-s − 0.256·61-s − 1.11·65-s + 1.34·67-s + 0.356·71-s − 0.936·73-s − 1.02·77-s − 1.68·79-s − 1.20·83-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  =  0
Selberg data  =  $(2,\ 10008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.789866521$
$L(\frac12)$  $\approx$  $2.789866521$
$L(\frac{3}{2})$   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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−16.84703725228209, −16.15337988505745, −15.55371598080497, −14.77451656860655, −14.20469028622130, −13.88739580394535, −13.20776208312931, −12.46322357295625, −12.32056596486721, −11.43622855966446, −10.55963590011573, −9.933164848153037, −9.738227156076473, −9.067479277907149, −8.550166815213094, −7.295561160279470, −7.002408708029717, −6.297084555855938, −5.563491547339970, −5.213930654872984, −4.125977816698797, −3.167685524650772, −2.761814625652164, −1.650504965322161, −0.8277216742030487, 0.8277216742030487, 1.650504965322161, 2.761814625652164, 3.167685524650772, 4.125977816698797, 5.213930654872984, 5.563491547339970, 6.297084555855938, 7.002408708029717, 7.295561160279470, 8.550166815213094, 9.067479277907149, 9.738227156076473, 9.933164848153037, 10.55963590011573, 11.43622855966446, 12.32056596486721, 12.46322357295625, 13.20776208312931, 13.88739580394535, 14.20469028622130, 14.77451656860655, 15.55371598080497, 16.15337988505745, 16.84703725228209

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