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Properties

Label 12-3120e6-1.1-c1e6-0-3
Degree 1212
Conductor 9.224×10209.224\times 10^{20}
Sign 11
Analytic cond. 2.39105×1082.39105\times 10^{8}
Root an. cond. 4.991324.99132
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 12·11-s − 4·19-s − 5·25-s + 20·29-s − 24·31-s − 20·41-s + 42·49-s − 12·59-s + 4·61-s − 16·71-s + 40·79-s + 6·81-s + 44·89-s − 36·99-s − 60·101-s + 24·109-s + 38·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 9-s + 3.61·11-s − 0.917·19-s − 25-s + 3.71·29-s − 4.31·31-s − 3.12·41-s + 6·49-s − 1.56·59-s + 0.512·61-s − 1.89·71-s + 4.50·79-s + 2/3·81-s + 4.66·89-s − 3.61·99-s − 5.97·101-s + 2.29·109-s + 3.45·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=((224⋅36⋅56⋅136)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅36⋅56⋅136)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 224⋅36⋅56⋅1362^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}
Sign: 11
Analytic conductor: 2.39105×1082.39105\times 10^{8}
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 224⋅36⋅56⋅136, ( :[1/2]6), 1)(12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 13.0852483513.08524835
L(12)L(\frac12) ≈\approx 13.0852483513.08524835
L(32)L(\frac{3}{2}) 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)
bad2 1 1
3 (1+T2)3 ( 1 + T^{2} )^{3}
5 1+pT2+8T3+p2T4+p3T6 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6}
13 (1+T2)3 ( 1 + T^{2} )^{3}
good7 (1−pT2)6 ( 1 - p T^{2} )^{6}
11 (1−6T+35T2−112T3+35pT4−6p2T5+p3T6)2 ( 1 - 6 T + 35 T^{2} - 112 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
17 1−46T2+1439T4−28068T6+1439p2T8−46p4T10+p6T12 1 - 46 T^{2} + 1439 T^{4} - 28068 T^{6} + 1439 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12}
19 (1+2T+33T2+92T3+33pT4+2p2T5+p3T6)2 ( 1 + 2 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 1−54T2+1871T4−46868T6+1871p2T8−54p4T10+p6T12 1 - 54 T^{2} + 1871 T^{4} - 46868 T^{6} + 1871 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12}
29 (1−10T+43T2−108T3+43pT4−10p2T5+p3T6)2 ( 1 - 10 T + 43 T^{2} - 108 T^{3} + 43 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 (1+12T+101T2+584T3+101pT4+12p2T5+p3T6)2 ( 1 + 12 T + 101 T^{2} + 584 T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1−130T2+9335T4−417660T6+9335p2T8−130p4T10+p6T12 1 - 130 T^{2} + 9335 T^{4} - 417660 T^{6} + 9335 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12}
41 (1+10T+89T2+488T3+89pT4+10p2T5+p3T6)2 ( 1 + 10 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 1−202T2+19015T4−1043212T6+19015p2T8−202p4T10+p6T12 1 - 202 T^{2} + 19015 T^{4} - 1043212 T^{6} + 19015 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12}
47 1−2pT2+7139T4−397884T6+7139p2T8−2p5T10+p6T12 1 - 2 p T^{2} + 7139 T^{4} - 397884 T^{6} + 7139 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12}
53 1−130T2+13655T4−792060T6+13655p2T8−130p4T10+p6T12 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12}
59 (1+6T+99T2+752T3+99pT4+6p2T5+p3T6)2 ( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1−2T+151T2−212T3+151pT4−2p2T5+p3T6)2 ( 1 - 2 T + 151 T^{2} - 212 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 1−130T2+12871T4−1093564T6+12871p2T8−130p4T10+p6T12 1 - 130 T^{2} + 12871 T^{4} - 1093564 T^{6} + 12871 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12}
71 (1+8T+215T2+1072T3+215pT4+8p2T5+p3T6)2 ( 1 + 8 T + 215 T^{2} + 1072 T^{3} + 215 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−106T2+14783T4−1127628T6+14783p2T8−106p4T10+p6T12 1 - 106 T^{2} + 14783 T^{4} - 1127628 T^{6} + 14783 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12}
79 (1−20T+345T2−3320T3+345pT4−20p2T5+p3T6)2 ( 1 - 20 T + 345 T^{2} - 3320 T^{3} + 345 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1−358T2+62555T4−6541356T6+62555p2T8−358p4T10+p6T12 1 - 358 T^{2} + 62555 T^{4} - 6541356 T^{6} + 62555 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12}
89 (1−22T+361T2−3672T3+361pT4−22p2T5+p3T6)2 ( 1 - 22 T + 361 T^{2} - 3672 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 (1−158T2+p2T4)3 ( 1 - 158 T^{2} + p^{2} T^{4} )^{3}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.58876805687438348338886408060, −4.22658777644963277224711219020, −4.07280584953882041810949398383, −3.88912970204085948270739178323, −3.84284258198109916668165331117, −3.77789327831133070368561813568, −3.77085828579888510112810546576, −3.56479881321858100176329149335, −3.55415593881964060690529633781, −3.02697237492869481377911624690, −2.99047286123964886112421061733, −2.90570967842718281498014123252, −2.61763409108892445073203585862, −2.45703590356028158849151447077, −2.43635022999084056155742735816, −2.14746807883972408889166258360, −1.79708542088910533601733190947, −1.65211422586707982345360402373, −1.64811885978877486401605361282, −1.47437469455835327633122598897, −1.45748957006103370277085726975, −0.71317089395745856639903613525, −0.69706277352882374359123859339, −0.57406262764946145722890282107, −0.43442245076966229069987308226, 0.43442245076966229069987308226, 0.57406262764946145722890282107, 0.69706277352882374359123859339, 0.71317089395745856639903613525, 1.45748957006103370277085726975, 1.47437469455835327633122598897, 1.64811885978877486401605361282, 1.65211422586707982345360402373, 1.79708542088910533601733190947, 2.14746807883972408889166258360, 2.43635022999084056155742735816, 2.45703590356028158849151447077, 2.61763409108892445073203585862, 2.90570967842718281498014123252, 2.99047286123964886112421061733, 3.02697237492869481377911624690, 3.55415593881964060690529633781, 3.56479881321858100176329149335, 3.77085828579888510112810546576, 3.77789327831133070368561813568, 3.84284258198109916668165331117, 3.88912970204085948270739178323, 4.07280584953882041810949398383, 4.22658777644963277224711219020, 4.58876805687438348338886408060

Graph of the ZZ-function along the critical line

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