Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 463 $
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 + 7-s + 6·11-s − 4·13-s − 19-s − 4·23-s + 4·25-s − 6·29-s + 3·35-s + 12·37-s + 6·41-s − 4·43-s + 9·47-s − 6·49-s − 5·53-s + 18·55-s − 61-s − 12·65-s − 10·67-s + 12·71-s + 11·73-s + 6·77-s − 12·79-s + 4·83-s + 14·89-s − 4·91-s − 3·95-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s + 1.80·11-s − 1.10·13-s − 0.229·19-s − 0.834·23-s + 4/5·25-s − 1.11·29-s + 0.507·35-s + 1.97·37-s + 0.937·41-s − 0.609·43-s + 1.31·47-s − 6/7·49-s − 0.686·53-s + 2.42·55-s − 0.128·61-s − 1.48·65-s − 1.22·67-s + 1.42·71-s + 1.28·73-s + 0.683·77-s − 1.35·79-s + 0.439·83-s + 1.48·89-s − 0.419·91-s − 0.307·95-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100008\)    =    \(2^{3} \cdot 3^{3} \cdot 463\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100008} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 100008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.108094184$
$L(\frac12)$  $\approx$  $4.108094184$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;463\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;463\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
463 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.92683158406621, −13.20003238342650, −12.92425006814079, −12.13194160612304, −11.99060093585636, −11.16297367426788, −10.99073613713494, −10.06678404028831, −9.728935354269436, −9.411120511029738, −8.994392290237131, −8.345645733678918, −7.605480183479838, −7.274887362940677, −6.494796000545555, −6.047222888415070, −5.851165955171045, −4.945159396131749, −4.538266225606619, −3.923961420390875, −3.267042387652466, −2.301394648248826, −2.067761058234593, −1.388671168870964, −0.6393248483572316, 0.6393248483572316, 1.388671168870964, 2.067761058234593, 2.301394648248826, 3.267042387652466, 3.923961420390875, 4.538266225606619, 4.945159396131749, 5.851165955171045, 6.047222888415070, 6.494796000545555, 7.274887362940677, 7.605480183479838, 8.345645733678918, 8.994392290237131, 9.411120511029738, 9.728935354269436, 10.06678404028831, 10.99073613713494, 11.16297367426788, 11.99060093585636, 12.13194160612304, 12.92425006814079, 13.20003238342650, 13.92683158406621

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