Properties

Label 4-1001e2-1.1-c1e2-0-6
Degree $4$
Conductor $1002001$
Sign $1$
Analytic cond. $63.8884$
Root an. cond. $2.82719$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 2·7-s − 5·9-s − 2·11-s + 12·16-s + 14·23-s − 9·25-s + 8·28-s − 4·29-s + 20·36-s − 22·37-s − 8·43-s + 8·44-s − 3·49-s + 4·53-s + 10·63-s − 32·64-s − 2·67-s − 18·71-s + 4·77-s + 16·79-s + 16·81-s − 56·92-s + 10·99-s + 36·100-s + 16·107-s + 8·109-s + ⋯
L(s)  = 1  − 2·4-s − 0.755·7-s − 5/3·9-s − 0.603·11-s + 3·16-s + 2.91·23-s − 9/5·25-s + 1.51·28-s − 0.742·29-s + 10/3·36-s − 3.61·37-s − 1.21·43-s + 1.20·44-s − 3/7·49-s + 0.549·53-s + 1.25·63-s − 4·64-s − 0.244·67-s − 2.13·71-s + 0.455·77-s + 1.80·79-s + 16/9·81-s − 5.83·92-s + 1.00·99-s + 18/5·100-s + 1.54·107-s + 0.766·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1002001\)    =    \(7^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(63.8884\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1002001,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.85612481771897939782236601586, −7.21661224338768342466420756206, −6.96627157214221326686940120925, −6.05777360202583450452013331209, −5.82135849339387357739621546932, −5.31104352782074944000454805478, −4.95237769114385482752321677072, −4.75768041533452306802121891710, −3.70099161938712630953878841304, −3.35775542055244311804083250514, −3.29304594675951575785139026233, −2.34448923217473262650057367432, −1.28655472231449922500586089553, 0, 0, 1.28655472231449922500586089553, 2.34448923217473262650057367432, 3.29304594675951575785139026233, 3.35775542055244311804083250514, 3.70099161938712630953878841304, 4.75768041533452306802121891710, 4.95237769114385482752321677072, 5.31104352782074944000454805478, 5.82135849339387357739621546932, 6.05777360202583450452013331209, 6.96627157214221326686940120925, 7.21661224338768342466420756206, 7.85612481771897939782236601586

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