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Properties

Label 20-847e10-1.1-c0e10-0-0
Degree 2020
Conductor 1.900×10291.900\times 10^{29}
Sign 11
Analytic cond. 0.0001821400.000182140
Root an. cond. 0.6501600.650160
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  − 2·2-s + 4-s − 7-s + 10·9-s − 11-s + 2·14-s − 20·18-s + 2·22-s + 9·23-s − 25-s − 28-s − 2·29-s + 10·36-s − 2·37-s − 2·43-s − 44-s − 18·46-s + 2·50-s − 2·53-s + 4·58-s − 10·63-s − 2·67-s − 2·71-s + 4·74-s + 77-s − 2·79-s + 55·81-s + ⋯
L(s)  = 1  − 2·2-s + 4-s − 7-s + 10·9-s − 11-s + 2·14-s − 20·18-s + 2·22-s + 9·23-s − 25-s − 28-s − 2·29-s + 10·36-s − 2·37-s − 2·43-s − 44-s − 18·46-s + 2·50-s − 2·53-s + 4·58-s − 10·63-s − 2·67-s − 2·71-s + 4·74-s + 77-s − 2·79-s + 55·81-s + ⋯

Functional equation

Λ(s)=((710⋅1120)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((710⋅1120)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 710⋅11207^{10} \cdot 11^{20}
Sign: 11
Analytic conductor: 0.0001821400.000182140
Root analytic conductor: 0.6501600.650160
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 710⋅1120, ( :[0]10), 1)(20,\ 7^{10} \cdot 11^{20} ,\ ( \ : [0]^{10} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.34707772680.3470777268
L(12)L(\frac12) ≈\approx 0.34707772680.3470777268
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)
bad7 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}
11 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}
good2 (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}
3 (1−T)10(1+T)10 ( 1 - T )^{10}( 1 + T )^{10}
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} )
13 (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} )
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} )
23 (1−T)10(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10) ( 1 - 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+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}
31 (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} )
37 (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}
41 (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} )
43 (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}
47 (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} )
53 (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}
59 (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} )
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)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+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}
73 (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} )
79 (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}
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

−3.95929769460795635013533977062, −3.87646094127987695373545787771, −3.86821169394955749031194947281, −3.68154530721578523172607219704, −3.63247732130412124206802493590, −3.44748069306915787434706461997, −3.40213915859692228723671761040, −3.25452678121079886770695943691, −3.15072789466233822118957036669, −3.03128453526091516159468638464, −2.87329574698440095347916999708, −2.71046856899198087982673694632, −2.49329779713884158149694995665, −2.32626641701206885927118657973, −2.28901676368649776640292863320, −2.25587779715616670595027667891, −1.66386079862212248245745449601, −1.62695862705032066453783647584, −1.55762667507813373628196350623, −1.54912102598778970747279317216, −1.35661443771431512359338153740, −1.32885188083122373874004572421, −1.23410552624123066304982293148, −1.02340311962395258080551571696, −0.942693466188934649593596428753, 0.942693466188934649593596428753, 1.02340311962395258080551571696, 1.23410552624123066304982293148, 1.32885188083122373874004572421, 1.35661443771431512359338153740, 1.54912102598778970747279317216, 1.55762667507813373628196350623, 1.62695862705032066453783647584, 1.66386079862212248245745449601, 2.25587779715616670595027667891, 2.28901676368649776640292863320, 2.32626641701206885927118657973, 2.49329779713884158149694995665, 2.71046856899198087982673694632, 2.87329574698440095347916999708, 3.03128453526091516159468638464, 3.15072789466233822118957036669, 3.25452678121079886770695943691, 3.40213915859692228723671761040, 3.44748069306915787434706461997, 3.63247732130412124206802493590, 3.68154530721578523172607219704, 3.86821169394955749031194947281, 3.87646094127987695373545787771, 3.95929769460795635013533977062

Graph of the ZZ-function along the critical line

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