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Properties

Label 20-603e10-1.1-c0e10-0-0
Degree 2020
Conductor 6.356×10276.356\times 10^{27}
Sign 11
Analytic cond. 6.09178×10−66.09178\times 10^{-6}
Root an. cond. 0.5485760.548576
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  − 4-s + 2·19-s − 25-s + 2·37-s − 49-s + 67-s − 9·73-s − 2·76-s − 11·79-s + 100-s + 2·103-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 2·19-s − 25-s + 2·37-s − 49-s + 67-s − 9·73-s − 2·76-s − 11·79-s + 100-s + 2·103-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

Λ(s)=((320⋅6710)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((320⋅6710)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 320⋅67103^{20} \cdot 67^{10}
Sign: 11
Analytic conductor: 6.09178×10−66.09178\times 10^{-6}
Root analytic conductor: 0.5485760.548576
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 320⋅6710, ( :[0]10), 1)(20,\ 3^{20} \cdot 67^{10} ,\ ( \ : [0]^{10} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.21506221030.2150622103
L(12)L(\frac12) ≈\approx 0.21506221030.2150622103
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 1
67 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)(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} )
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+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−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}
19 (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}
23 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)10 ( 1 + T^{2} )^{10}
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)(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−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}
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−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}
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} )
71 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}
73 (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} )
79 (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} )
83 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}
89 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}
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

−4.27435616543027688599293677642, −4.21715688595092454354017600332, −4.12594544828655133336517240552, −4.03424171486995322608799756395, −3.96290099593183505446946579415, −3.68242528829453880750944554046, −3.54706420153946336799167714220, −3.33836695661678577911828177385, −3.21771539877576185127359244041, −3.17548301210851036533894362356, −2.94473655255086045954646538007, −2.87802444583166537838806422201, −2.84685167386361119024353477069, −2.76516195765349480175014134749, −2.75672644026665761760217654210, −2.57711630219261795855528040253, −2.32469343134141155024194326993, −1.86849589453599506482216752853, −1.75242890150317825559568806170, −1.75076773683840588523257063324, −1.58868602312400784293260916095, −1.58094450702948420210521010289, −1.29179929140820329567421659301, −1.08167113336342232021338488685, −0.69892488463632954080049801154, 0.69892488463632954080049801154, 1.08167113336342232021338488685, 1.29179929140820329567421659301, 1.58094450702948420210521010289, 1.58868602312400784293260916095, 1.75076773683840588523257063324, 1.75242890150317825559568806170, 1.86849589453599506482216752853, 2.32469343134141155024194326993, 2.57711630219261795855528040253, 2.75672644026665761760217654210, 2.76516195765349480175014134749, 2.84685167386361119024353477069, 2.87802444583166537838806422201, 2.94473655255086045954646538007, 3.17548301210851036533894362356, 3.21771539877576185127359244041, 3.33836695661678577911828177385, 3.54706420153946336799167714220, 3.68242528829453880750944554046, 3.96290099593183505446946579415, 4.03424171486995322608799756395, 4.12594544828655133336517240552, 4.21715688595092454354017600332, 4.27435616543027688599293677642

Graph of the ZZ-function along the critical line

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