Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 463 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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  =  1
Selberg data  =  $(2,\ 100008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;463\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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

−14.06190031360102, −13.35304244467279, −12.89298449369798, −12.67290417287411, −11.92443967983242, −11.53795724930566, −11.22135367714988, −10.54710208104922, −10.10395260226906, −9.748866049090142, −8.821324517157341, −8.405448999661298, −7.925813364854063, −7.586125398245377, −7.168193452776972, −6.515317426190602, −5.776823466512263, −5.072106860887832, −4.683766353034323, −4.423875543696565, −3.446430094750294, −2.906091933652088, −2.536784905006263, −1.622567650048351, −0.6095319619899024, 0, 0.6095319619899024, 1.622567650048351, 2.536784905006263, 2.906091933652088, 3.446430094750294, 4.423875543696565, 4.683766353034323, 5.072106860887832, 5.776823466512263, 6.515317426190602, 7.168193452776972, 7.586125398245377, 7.925813364854063, 8.405448999661298, 8.821324517157341, 9.748866049090142, 10.10395260226906, 10.54710208104922, 11.22135367714988, 11.53795724930566, 11.92443967983242, 12.67290417287411, 12.89298449369798, 13.35304244467279, 14.06190031360102

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