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Properties

Label 20-253e10-1.1-c0e10-0-0
Degree 2020
Conductor 1.074×10241.074\times 10^{24}
Sign 11
Analytic cond. 1.02985×10−91.02985\times 10^{-9}
Root an. cond. 0.3553350.355335
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  − 2·3-s − 4-s − 2·5-s + 9-s − 11-s + 2·12-s + 4·15-s + 2·20-s − 23-s + 25-s − 2·31-s + 2·33-s − 36-s − 2·37-s + 44-s − 2·45-s − 2·47-s − 49-s − 2·53-s + 2·55-s − 2·59-s − 4·60-s − 2·67-s + 2·69-s + 9·71-s − 2·75-s − 2·89-s + ⋯
L(s)  = 1  − 2·3-s − 4-s − 2·5-s + 9-s − 11-s + 2·12-s + 4·15-s + 2·20-s − 23-s + 25-s − 2·31-s + 2·33-s − 36-s − 2·37-s + 44-s − 2·45-s − 2·47-s − 49-s − 2·53-s + 2·55-s − 2·59-s − 4·60-s − 2·67-s + 2·69-s + 9·71-s − 2·75-s − 2·89-s + ⋯

Functional equation

Λ(s)=((1110⋅2310)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((1110⋅2310)s/2 ΓC(s)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(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: 1110⋅231011^{10} \cdot 23^{10}
Sign: 11
Analytic conductor: 1.02985×10−91.02985\times 10^{-9}
Root analytic conductor: 0.3553350.355335
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 1110⋅2310, ( :[0]10), 1)(20,\ 11^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.010219635740.01021963574
L(12)L(\frac12) ≈\approx 0.010219635740.01021963574
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)
bad11 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−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} )
3 (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}
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−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} )
29 (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} )
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)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+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}
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)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)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)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−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)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

−4.96332786324097638901111599888, −4.95535212339125014341763807347, −4.80806915706543327818837999032, −4.77476252338135007775150815745, −4.77194773003216484007251457075, −4.20033045365264577573635538334, −4.09732894344899986174028099083, −4.08620711099445895595878795897, −4.06621496211129415832596826603, −3.94106124337465679784161016062, −3.73837341827522809199844503466, −3.66906143112531900718428331295, −3.27037138492957112978123090684, −3.26744404599477569037837159736, −3.25297195246930043817121171869, −3.19252179404765312929147401789, −2.96486559833008781205357508615, −2.54329993479200130555116758660, −2.45557400861177700161023646592, −2.15046498995458974666359606139, −2.04958587117098386591449677648, −1.76931037847701579835770158532, −1.68272420117990294505165198837, −1.48027246511287223912638691342, −0.68990414883233159863030181307, 0.68990414883233159863030181307, 1.48027246511287223912638691342, 1.68272420117990294505165198837, 1.76931037847701579835770158532, 2.04958587117098386591449677648, 2.15046498995458974666359606139, 2.45557400861177700161023646592, 2.54329993479200130555116758660, 2.96486559833008781205357508615, 3.19252179404765312929147401789, 3.25297195246930043817121171869, 3.26744404599477569037837159736, 3.27037138492957112978123090684, 3.66906143112531900718428331295, 3.73837341827522809199844503466, 3.94106124337465679784161016062, 4.06621496211129415832596826603, 4.08620711099445895595878795897, 4.09732894344899986174028099083, 4.20033045365264577573635538334, 4.77194773003216484007251457075, 4.77476252338135007775150815745, 4.80806915706543327818837999032, 4.95535212339125014341763807347, 4.96332786324097638901111599888

Graph of the ZZ-function along the critical line

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