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Properties

Label 20-23e20-1.1-c0e10-0-0
Degree 2020
Conductor 1.716×10271.716\times 10^{27}
Sign 11
Analytic cond. 1.64485×10−61.64485\times 10^{-6}
Root an. cond. 0.5138140.513814
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s + 13-s + 18-s − 25-s + 26-s + 29-s + 31-s + 36-s + 39-s + 41-s − 10·47-s − 49-s − 50-s + 52-s + 58-s − 2·59-s + 62-s + 71-s + 73-s − 75-s + 78-s + 82-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s + 13-s + 18-s − 25-s + 26-s + 29-s + 31-s + 36-s + 39-s + 41-s − 10·47-s − 49-s − 50-s + 52-s + 58-s − 2·59-s + 62-s + 71-s + 73-s − 75-s + 78-s + 82-s + ⋯

Functional equation

Λ(s)=((2320)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((2320)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 232023^{20}
Sign: 11
Analytic conductor: 1.64485×10−61.64485\times 10^{-6}
Root analytic conductor: 0.5138140.513814
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 2320, ( :[0]10), 1)(20,\ 23^{20} ,\ ( \ : [0]^{10} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.68308163010.6830816301
L(12)L(\frac12) ≈\approx 0.68308163010.6830816301
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad23 1 1
good2 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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}
3 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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}
5 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
7 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
11 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
13 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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}
17 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
19 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
29 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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}
31 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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}
37 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
41 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
47 (1+T+T2)10 ( 1 + T + T^{2} )^{10}
53 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
59 (1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10)2 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}
61 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
67 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
71 1−T+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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+T3−T4+T6−T7+T9−T10+T11−T13+T14−T16+T17−T19+T20 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+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
83 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
89 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
97 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )
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   L(s)=∏p ∏j=120(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.31346669602735780501556666432, −4.31122786587855878910505728878, −4.25544739428256422348195987102, −4.11610886288602929012346213287, −3.73719927636632843619940681746, −3.50030186170233522560942329026, −3.48145704318226877760266580882, −3.47912672643543325040992832221, −3.45507613431955823868264944697, −3.38771423182477079091132231744, −3.24282800077695358780020091063, −3.19890610202039675015855632780, −2.91565639763028195984950920428, −2.78351403199132428138375055544, −2.53146292675253785001350500437, −2.46210895350047538355852698316, −2.41459347363938636346834332744, −2.39033656988996553716550406438, −1.98832976613431978869891846814, −1.90660409659246277446888356648, −1.61711021586480188255228548776, −1.49090835484882860715703639907, −1.37860327792291508325272120362, −1.33377017719317223005252703081, −1.18544133226214076330003986811, 1.18544133226214076330003986811, 1.33377017719317223005252703081, 1.37860327792291508325272120362, 1.49090835484882860715703639907, 1.61711021586480188255228548776, 1.90660409659246277446888356648, 1.98832976613431978869891846814, 2.39033656988996553716550406438, 2.41459347363938636346834332744, 2.46210895350047538355852698316, 2.53146292675253785001350500437, 2.78351403199132428138375055544, 2.91565639763028195984950920428, 3.19890610202039675015855632780, 3.24282800077695358780020091063, 3.38771423182477079091132231744, 3.45507613431955823868264944697, 3.47912672643543325040992832221, 3.48145704318226877760266580882, 3.50030186170233522560942329026, 3.73719927636632843619940681746, 4.11610886288602929012346213287, 4.25544739428256422348195987102, 4.31122786587855878910505728878, 4.31346669602735780501556666432

Graph of the ZZ-function along the critical line

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