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Properties

Label 40-1288e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.579\times 10^{62}$
Sign $1$
Analytic cond. $0.000145030$
Root an. cond. $0.801745$
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  + 2·2-s + 4-s + 2·7-s − 9·9-s + 4·14-s − 18·18-s − 2·23-s + 2·25-s + 2·28-s − 9·36-s − 4·46-s + 49-s + 4·50-s − 18·63-s − 18·71-s + 4·79-s + 45·81-s − 2·92-s + 2·98-s + 2·100-s + 4·113-s − 2·121-s − 36·126-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2·2-s + 4-s + 2·7-s − 9·9-s + 4·14-s − 18·18-s − 2·23-s + 2·25-s + 2·28-s − 9·36-s − 4·46-s + 49-s + 4·50-s − 18·63-s − 18·71-s + 4·79-s + 45·81-s − 2·92-s + 2·98-s + 2·100-s + 4·113-s − 2·121-s − 36·126-s + 127-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{60} \cdot 7^{20} \cdot 23^{20}\)
Sign: $1$
Analytic conductor: \(0.000145030\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{60} \cdot 7^{20} \cdot 23^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2253022712\)
\(L(\frac12)\) \(\approx\) \(0.2253022712\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
good3 \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
47 \( ( 1 - T )^{20}( 1 + T )^{20} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
71 \( ( 1 + T )^{20}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{4} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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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

−2.52110880840036564651764683557, −2.37589550201853217347796073456, −2.36925230612131970056180828466, −2.31191268590217128527486918632, −2.22514373369760617176568979740, −2.14429910273445274426348221587, −2.03886146432110274231945961839, −1.92172441887968405579905805951, −1.85037519932057185299678792105, −1.79723509494951313722819496970, −1.77584620558823344072405411188, −1.68424333045715390627349916225, −1.65307003948230781842473856229, −1.64328815021791284846094714116, −1.55110313413638505123345766043, −1.32945989599940289912900892939, −1.30576748683286032218510630493, −1.28461879307519293115988975075, −1.18314021850846816130818321708, −0.982137622351988802136592780138, −0.952729328405093493901870066743, −0.929414086801593434536903366374, −0.47965759687002671942515294071, −0.45960225095452461951426694721, −0.31945908548946033373750280925, 0.31945908548946033373750280925, 0.45960225095452461951426694721, 0.47965759687002671942515294071, 0.929414086801593434536903366374, 0.952729328405093493901870066743, 0.982137622351988802136592780138, 1.18314021850846816130818321708, 1.28461879307519293115988975075, 1.30576748683286032218510630493, 1.32945989599940289912900892939, 1.55110313413638505123345766043, 1.64328815021791284846094714116, 1.65307003948230781842473856229, 1.68424333045715390627349916225, 1.77584620558823344072405411188, 1.79723509494951313722819496970, 1.85037519932057185299678792105, 1.92172441887968405579905805951, 2.03886146432110274231945961839, 2.14429910273445274426348221587, 2.22514373369760617176568979740, 2.31191268590217128527486918632, 2.36925230612131970056180828466, 2.37589550201853217347796073456, 2.52110880840036564651764683557

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

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