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Properties

Label 40-1407e20-1.1-c0e20-0-2
Degree $40$
Conductor $9.244\times 10^{62}$
Sign $1$
Analytic cond. $0.000849228$
Root an. cond. $0.837964$
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  + 3-s + 2·4-s − 7-s + 9-s + 2·12-s + 2·13-s + 16-s − 21-s + 25-s − 2·28-s + 4·31-s + 2·36-s + 2·37-s + 2·39-s + 48-s + 49-s + 4·52-s − 2·61-s − 63-s − 67-s − 8·73-s + 75-s + 8·79-s − 2·84-s − 2·91-s + 4·93-s + 97-s + ⋯
L(s)  = 1  + 3-s + 2·4-s − 7-s + 9-s + 2·12-s + 2·13-s + 16-s − 21-s + 25-s − 2·28-s + 4·31-s + 2·36-s + 2·37-s + 2·39-s + 48-s + 49-s + 4·52-s − 2·61-s − 63-s − 67-s − 8·73-s + 75-s + 8·79-s − 2·84-s − 2·91-s + 4·93-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{20} \cdot 7^{20} \cdot 67^{20}\)
Sign: $1$
Analytic conductor: \(0.000849228\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{20} \cdot 7^{20} \cdot 67^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.527661001\)
\(L(\frac12)\) \(\approx\) \(2.527661001\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
7 \( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} \)
67 \( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} \)
good2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
17 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
29 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{4} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
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 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
53 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
59 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
61 \( ( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} )^{2} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
73 \( ( 1 + T + T^{2} )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
79 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
83 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
89 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
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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

−2.31771085157682092348250209088, −2.24251869417704563757494318170, −2.19836728086716354875037631179, −2.14449060197038141253772935372, −2.11980221493369283044055117085, −2.11696056077434657467223757650, −2.01701578510686864303298177414, −1.96311762615408353584396952253, −1.90759502706131066113973381323, −1.78571088385958187849219099854, −1.72166448917007778309369517735, −1.69128082657731859178679967565, −1.61995979299475932804378228882, −1.33776093967430327659200607525, −1.30306125832757166067418112998, −1.28466571490235707264091840788, −1.23612880282652823563804675053, −1.22831713221134968030026347555, −1.15818963831590445725301818884, −1.11932982030622035772854814345, −1.02355783586509127799310358180, −0.994661365773624088112005050129, −0.899011241273848017558106844292, −0.62502996856896733153832024092, −0.44008669005818252623911333446, 0.44008669005818252623911333446, 0.62502996856896733153832024092, 0.899011241273848017558106844292, 0.994661365773624088112005050129, 1.02355783586509127799310358180, 1.11932982030622035772854814345, 1.15818963831590445725301818884, 1.22831713221134968030026347555, 1.23612880282652823563804675053, 1.28466571490235707264091840788, 1.30306125832757166067418112998, 1.33776093967430327659200607525, 1.61995979299475932804378228882, 1.69128082657731859178679967565, 1.72166448917007778309369517735, 1.78571088385958187849219099854, 1.90759502706131066113973381323, 1.96311762615408353584396952253, 2.01701578510686864303298177414, 2.11696056077434657467223757650, 2.11980221493369283044055117085, 2.14449060197038141253772935372, 2.19836728086716354875037631179, 2.24251869417704563757494318170, 2.31771085157682092348250209088

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

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