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Properties

Label 40-869e20-1.1-c0e20-0-0
Degree $40$
Conductor $6.031\times 10^{58}$
Sign $1$
Analytic cond. $5.54036\times 10^{-8}$
Root an. cond. $0.658549$
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  − 5·9-s − 2·32-s − 5·49-s − 5·79-s + 10·81-s + 15·83-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 + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 5·9-s − 2·32-s − 5·49-s − 5·79-s + 10·81-s + 15·83-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 + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 79^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 79^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(11^{20} \cdot 79^{20}\)
Sign: $1$
Analytic conductor: \(5.54036\times 10^{-8}\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 11^{20} \cdot 79^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003164529863\)
\(L(\frac12)\) \(\approx\) \(0.0003164529863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
good2 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
5 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
13 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
19 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
23 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
31 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
43 \( ( 1 - T )^{20}( 1 + T )^{20} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
67 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
73 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
83 \( ( 1 - T )^{20}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
89 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
97 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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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

−2.61432916058158973326059900528, −2.60527746224242096665925153334, −2.48902057699292373084761570769, −2.37418355369437312437999438159, −2.27147826417288976803704682619, −2.26107167974162340489339000407, −2.19890560075597474813551305929, −2.13345714951986106479017124540, −2.08493534200915078985676191043, −2.05045027916809333370138674364, −1.98258010952793066961948010652, −1.77787488811084348890061319681, −1.74414356698575704406689491610, −1.68405857144169826133496392287, −1.66998238131836381551886017037, −1.60163181060002706234892674355, −1.59067977021499034058902461662, −1.15179391303695001800305845858, −1.12987056450597520703206116259, −1.11424292704568100250996511998, −1.04080633391758185682198524141, −1.03889774240273580466912475836, −0.889662271639472934932589881810, −0.802447250753115764386888569728, −0.01947391187140906497996998874, 0.01947391187140906497996998874, 0.802447250753115764386888569728, 0.889662271639472934932589881810, 1.03889774240273580466912475836, 1.04080633391758185682198524141, 1.11424292704568100250996511998, 1.12987056450597520703206116259, 1.15179391303695001800305845858, 1.59067977021499034058902461662, 1.60163181060002706234892674355, 1.66998238131836381551886017037, 1.68405857144169826133496392287, 1.74414356698575704406689491610, 1.77787488811084348890061319681, 1.98258010952793066961948010652, 2.05045027916809333370138674364, 2.08493534200915078985676191043, 2.13345714951986106479017124540, 2.19890560075597474813551305929, 2.26107167974162340489339000407, 2.27147826417288976803704682619, 2.37418355369437312437999438159, 2.48902057699292373084761570769, 2.60527746224242096665925153334, 2.61432916058158973326059900528

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

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