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Properties

Label 20-759e10-1.1-c0e10-0-0
Degree 2020
Conductor 6.345×10286.345\times 10^{28}
Sign 11
Analytic cond. 6.08119×10−56.08119\times 10^{-5}
Root an. cond. 0.6154590.615459
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  + 3-s + 4-s − 2·5-s + 11-s + 12-s − 2·15-s − 2·20-s − 23-s + 25-s − 2·31-s + 33-s + 44-s + 49-s + 2·53-s − 2·55-s − 2·60-s − 69-s − 11·71-s + 75-s + 2·89-s − 92-s − 2·93-s − 11·97-s + 100-s + 2·113-s + 2·115-s − 2·124-s + ⋯
L(s)  = 1  + 3-s + 4-s − 2·5-s + 11-s + 12-s − 2·15-s − 2·20-s − 23-s + 25-s − 2·31-s + 33-s + 44-s + 49-s + 2·53-s − 2·55-s − 2·60-s − 69-s − 11·71-s + 75-s + 2·89-s − 92-s − 2·93-s − 11·97-s + 100-s + 2·113-s + 2·115-s − 2·124-s + ⋯

Functional equation

Λ(s)=((310⋅1110⋅2310)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((310⋅1110⋅2310)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 310⋅1110⋅23103^{10} \cdot 11^{10} \cdot 23^{10}
Sign: 11
Analytic conductor: 6.08119×10−56.08119\times 10^{-5}
Root analytic conductor: 0.6154590.615459
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 310⋅1110⋅2310, ( :[0]10), 1)(20,\ 3^{10} \cdot 11^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.42986266050.4298626605
L(12)L(\frac12) ≈\approx 0.42986266050.4298626605
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)
bad3 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}
23 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−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
5 (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}
7 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
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−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
29 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
31 (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}
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−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
43 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
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−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
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)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} )
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−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}
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)2 ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}
97 (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} )
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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

−4.13169368230537108057237909656, −4.04512912232849494973551336004, −3.86755288993258236671664488487, −3.78012310341311797455699975717, −3.67290301885061116941650794845, −3.66244933132070070713906348165, −3.30176179402836884952348962350, −3.10427308500204328268688528516, −3.01682155691003008089464237517, −3.01559389182059657115658802879, −3.01332629440400930167828346057, −2.91570743693391176177554578745, −2.68216118391603710036992846260, −2.66414727933795613182352424142, −2.64245273271469038423740182094, −2.31996272233947653938819050435, −2.05180864387079366975673841630, −1.82068606928102639317806062344, −1.76889651018108094712603290338, −1.68069007868343073209400674611, −1.66851182074270606768332203879, −1.57218751463851208188347919730, −1.34855651537476016162739000380, −0.843907767160929969591719817391, −0.64511802082160709157692683767, 0.64511802082160709157692683767, 0.843907767160929969591719817391, 1.34855651537476016162739000380, 1.57218751463851208188347919730, 1.66851182074270606768332203879, 1.68069007868343073209400674611, 1.76889651018108094712603290338, 1.82068606928102639317806062344, 2.05180864387079366975673841630, 2.31996272233947653938819050435, 2.64245273271469038423740182094, 2.66414727933795613182352424142, 2.68216118391603710036992846260, 2.91570743693391176177554578745, 3.01332629440400930167828346057, 3.01559389182059657115658802879, 3.01682155691003008089464237517, 3.10427308500204328268688528516, 3.30176179402836884952348962350, 3.66244933132070070713906348165, 3.67290301885061116941650794845, 3.78012310341311797455699975717, 3.86755288993258236671664488487, 4.04512912232849494973551336004, 4.13169368230537108057237909656

Graph of the ZZ-function along the critical line

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