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Properties

Label 40-1407e20-1.1-c0e20-0-3
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 − 4-s − 2·7-s + 9-s + 12-s − 2·13-s + 16-s + 3·19-s + 2·21-s + 25-s + 2·28-s + 2·31-s − 36-s − 37-s + 2·39-s − 48-s + 49-s + 2·52-s − 3·57-s − 61-s − 2·63-s + 2·67-s + 11·73-s − 75-s − 3·76-s + 11·79-s − 2·84-s + ⋯
L(s)  = 1  − 3-s − 4-s − 2·7-s + 9-s + 12-s − 2·13-s + 16-s + 3·19-s + 2·21-s + 25-s + 2·28-s + 2·31-s − 36-s − 37-s + 2·39-s − 48-s + 49-s + 2·52-s − 3·57-s − 61-s − 2·63-s + 2·67-s + 11·73-s − 75-s − 3·76-s + 11·79-s − 2·84-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4939688167\)
\(L(\frac12)\) \(\approx\) \(0.4939688167\)
\(L(1)\) 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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
good2 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
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 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \)
29 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
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 \( ( 1 - T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
79 \( ( 1 - T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
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

−2.33647211547592265910259641821, −2.26059645809018018219264242529, −2.16838607416288083456682158732, −2.16723592991673922730858451081, −2.12307633189289024427545966301, −2.01949919561065729356165337213, −1.93333754539138299617388778221, −1.92562660989603884291954105874, −1.91370092951444069283738025145, −1.87306280060931463221214693623, −1.83591964778193283485430241604, −1.81195888455005542476241577883, −1.53794047822811673940488186011, −1.37546578361237762968400389448, −1.19697645236610432906144764168, −1.16153499467736481530000822308, −1.11424581089159734210208830060, −1.04075624889804836612365351607, −1.01696196587351401433076820678, −0.950864833001833931104554822445, −0.948077764025661497540929577030, −0.811031524642687589974154605768, −0.791816411765763963408926329982, −0.62332212223731833816210926178, −0.50954482956888653242004690697, 0.50954482956888653242004690697, 0.62332212223731833816210926178, 0.791816411765763963408926329982, 0.811031524642687589974154605768, 0.948077764025661497540929577030, 0.950864833001833931104554822445, 1.01696196587351401433076820678, 1.04075624889804836612365351607, 1.11424581089159734210208830060, 1.16153499467736481530000822308, 1.19697645236610432906144764168, 1.37546578361237762968400389448, 1.53794047822811673940488186011, 1.81195888455005542476241577883, 1.83591964778193283485430241604, 1.87306280060931463221214693623, 1.91370092951444069283738025145, 1.92562660989603884291954105874, 1.93333754539138299617388778221, 2.01949919561065729356165337213, 2.12307633189289024427545966301, 2.16723592991673922730858451081, 2.16838607416288083456682158732, 2.26059645809018018219264242529, 2.33647211547592265910259641821

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

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