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Properties

Label 40-1057e20-1.1-c0e20-0-0
Degree $40$
Conductor $3.030\times 10^{60}$
Sign $1$
Analytic cond. $2.78382\times 10^{-6}$
Root an. cond. $0.726300$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·23-s − 5·29-s − 2·32-s − 5·53-s − 5·67-s − 5·71-s + 20·79-s − 5·107-s + 20·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 10·23-s − 5·29-s − 2·32-s − 5·53-s − 5·67-s − 5·71-s + 20·79-s − 5·107-s + 20·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(7^{20} \cdot 151^{20}\)
Sign: $1$
Analytic conductor: \(2.78382\times 10^{-6}\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 7^{20} \cdot 151^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04743950975\)
\(L(\frac12)\) \(\approx\) \(0.04743950975\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
151 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
good2 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
3 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
5 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
11 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
17 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{10} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
31 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
37 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
41 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
43 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
47 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
61 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
73 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
79 \( ( 1 - T )^{20}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
83 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
89 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
97 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.33169864192583890746400198383, −2.28743572359455333743551740817, −2.28583332046013511419683848682, −2.27676312889523746163624670677, −2.26922885970598280870882157539, −2.22137514310072611451146775072, −2.07116501234742544196504656380, −1.99789971438515719443656681889, −1.97357633652005767016980091786, −1.78582796368061324224836404072, −1.77222736802031993999997374507, −1.75646113122650439240349456618, −1.74918707017431859653826619995, −1.71571792009562035541334100186, −1.71119226171164130193883083888, −1.57042794674705679969418597088, −1.47069979630521444878229877821, −1.43825031950044974993650038233, −1.20311959245771802153182911921, −1.09083693421389469806344853908, −0.926493714703654115759421922422, −0.885138265542674246257109914469, −0.66637044363566623125739278889, −0.64950637308375359569075091734, −0.18712968701513081165436282069, 0.18712968701513081165436282069, 0.64950637308375359569075091734, 0.66637044363566623125739278889, 0.885138265542674246257109914469, 0.926493714703654115759421922422, 1.09083693421389469806344853908, 1.20311959245771802153182911921, 1.43825031950044974993650038233, 1.47069979630521444878229877821, 1.57042794674705679969418597088, 1.71119226171164130193883083888, 1.71571792009562035541334100186, 1.74918707017431859653826619995, 1.75646113122650439240349456618, 1.77222736802031993999997374507, 1.78582796368061324224836404072, 1.97357633652005767016980091786, 1.99789971438515719443656681889, 2.07116501234742544196504656380, 2.22137514310072611451146775072, 2.26922885970598280870882157539, 2.27676312889523746163624670677, 2.28583332046013511419683848682, 2.28743572359455333743551740817, 2.33169864192583890746400198383

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

Plot not available for L-functions of degree greater than 10.