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Properties

Label 40-2008e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.136\times 10^{66}$
Sign $1$
Analytic cond. $1.04331$
Root an. cond. $1.00106$
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  − 5·2-s + 10·4-s − 10·8-s − 5·11-s + 5·16-s + 25·22-s − 5·25-s + 4·32-s − 50·44-s + 25·50-s − 5·59-s − 25·64-s − 5·83-s + 50·88-s − 50·100-s + 20·107-s + 25·118-s + 10·121-s + 127-s + 50·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 5·2-s + 10·4-s − 10·8-s − 5·11-s + 5·16-s + 25·22-s − 5·25-s + 4·32-s − 50·44-s + 25·50-s − 5·59-s − 25·64-s − 5·83-s + 50·88-s − 50·100-s + 20·107-s + 25·118-s + 10·121-s + 127-s + 50·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 251^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 251^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{60} \cdot 251^{20}\)
Sign: $1$
Analytic conductor: \(1.04331\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{60} \cdot 251^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0008769381985\)
\(L(\frac12)\) \(\approx\) \(0.0008769381985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
251 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
good3 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
7 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
17 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
19 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
23 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
29 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
31 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
37 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
41 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
43 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
53 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
61 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
67 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
71 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
73 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
79 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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

−2.14015869448479888366962193428, −2.11581054596439252930185835349, −2.04747655476175728612447316343, −2.02894913013067076729499434238, −1.98569930727970895796731720164, −1.95922980099398377633177722277, −1.79803565395635955937090670251, −1.78968767762393015080956490071, −1.67508834049778948185072242269, −1.62619200283082121889960387032, −1.36200027358655626369629912297, −1.34637583584933336387511859175, −1.33814891504065739918745781922, −1.25954878001978811915229432244, −1.24483045587200497462688898506, −1.18991494936720331300159800725, −1.15022970780373857315297386509, −1.00011762086147921473333154331, −0.799574256329796414513515281778, −0.74244609273313889607080179121, −0.71484707269729334135984375663, −0.58311975489442036627797167741, −0.36637089423054416663441595441, −0.24745098228046676060577755577, −0.17238931423082641288108348808, 0.17238931423082641288108348808, 0.24745098228046676060577755577, 0.36637089423054416663441595441, 0.58311975489442036627797167741, 0.71484707269729334135984375663, 0.74244609273313889607080179121, 0.799574256329796414513515281778, 1.00011762086147921473333154331, 1.15022970780373857315297386509, 1.18991494936720331300159800725, 1.24483045587200497462688898506, 1.25954878001978811915229432244, 1.33814891504065739918745781922, 1.34637583584933336387511859175, 1.36200027358655626369629912297, 1.62619200283082121889960387032, 1.67508834049778948185072242269, 1.78968767762393015080956490071, 1.79803565395635955937090670251, 1.95922980099398377633177722277, 1.98569930727970895796731720164, 2.02894913013067076729499434238, 2.04747655476175728612447316343, 2.11581054596439252930185835349, 2.14015869448479888366962193428

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

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