Properties

Label 40-1151e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.665\times 10^{61}$
Sign $1$
Analytic cond. $1.52985\times 10^{-5}$
Root an. cond. $0.757907$
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-s − 3-s − 5-s + 6-s − 7-s + 10-s − 11-s + 14-s + 15-s + 21-s + 22-s − 29-s − 30-s + 33-s + 35-s − 37-s − 42-s − 43-s − 47-s − 53-s + 55-s + 58-s − 59-s − 66-s − 67-s − 70-s + 74-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s − 11-s + 14-s + 15-s + 21-s + 22-s − 29-s − 30-s + 33-s + 35-s − 37-s − 42-s − 43-s − 47-s − 53-s + 55-s + 58-s − 59-s − 66-s − 67-s − 70-s + 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(1151^{20}\)
Sign: $1$
Analytic conductor: \(1.52985\times 10^{-5}\)
Root analytic conductor: \(0.757907\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1151} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 1151^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04414772255\)
\(L(\frac12)\) \(\approx\) \(0.04414772255\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1151 \( ( 1 - T )^{20} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
13 \( ( 1 - T )^{20}( 1 + T )^{20} \)
17 \( ( 1 - T )^{20}( 1 + T )^{20} \)
19 \( ( 1 - T )^{20}( 1 + T )^{20} \)
23 \( ( 1 - T )^{20}( 1 + T )^{20} \)
29 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
31 \( ( 1 - T )^{20}( 1 + T )^{20} \)
37 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
41 \( ( 1 - T )^{20}( 1 + T )^{20} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
47 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
61 \( ( 1 - T )^{20}( 1 + T )^{20} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
71 \( ( 1 - T )^{20}( 1 + T )^{20} \)
73 \( ( 1 - T )^{20}( 1 + T )^{20} \)
79 \( ( 1 - T )^{20}( 1 + T )^{20} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} + T^{37} + T^{38} + T^{39} + T^{40} \)
89 \( ( 1 - T )^{20}( 1 + T )^{20} \)
97 \( ( 1 - T )^{20}( 1 + T )^{20} \)
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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

−2.47997188640276247669387581233, −2.39955535016270297672760146506, −2.34056937364751915387674178229, −2.27867219134959477698305717683, −2.12685889501179184621898839212, −2.05648194940788566989001570881, −1.99122131413727084432944259024, −1.97597329520505202385499367031, −1.77519154902324933320564370563, −1.74418135004610338397050114911, −1.72878795897856266977179637549, −1.67491823708135153088302467569, −1.64122798542492892300041644734, −1.57708463105678981137874171039, −1.54703238992230805996035204824, −1.46592678287770365278785258856, −1.41911301658151367845752220256, −1.21516518197759233868262899195, −1.12855445454651058013647277599, −0.936941154694994616074170576293, −0.920359813256037696871152919438, −0.70086523737606361087299374895, −0.60545956681672527135147983089, −0.51359206825369711854163410691, −0.39974765951228394358433974699, 0.39974765951228394358433974699, 0.51359206825369711854163410691, 0.60545956681672527135147983089, 0.70086523737606361087299374895, 0.920359813256037696871152919438, 0.936941154694994616074170576293, 1.12855445454651058013647277599, 1.21516518197759233868262899195, 1.41911301658151367845752220256, 1.46592678287770365278785258856, 1.54703238992230805996035204824, 1.57708463105678981137874171039, 1.64122798542492892300041644734, 1.67491823708135153088302467569, 1.72878795897856266977179637549, 1.74418135004610338397050114911, 1.77519154902324933320564370563, 1.97597329520505202385499367031, 1.99122131413727084432944259024, 2.05648194940788566989001570881, 2.12685889501179184621898839212, 2.27867219134959477698305717683, 2.34056937364751915387674178229, 2.39955535016270297672760146506, 2.47997188640276247669387581233

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

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