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Properties

Label 12-1560e6-1.1-c1e6-0-1
Degree 1212
Conductor 1.441×10191.441\times 10^{19}
Sign 11
Analytic cond. 3.73602×1063.73602\times 10^{6}
Root an. cond. 3.529393.52939
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)=((218⋅36⋅56⋅136)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \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)=((218⋅36⋅56⋅136)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \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: 218⋅36⋅56⋅1362^{18} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}
Sign: 11
Analytic conductor: 3.73602×1063.73602\times 10^{6}
Root analytic conductor: 3.529393.52939
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅36⋅56⋅136, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.5497854692.549785469
L(12)L(\frac12) ≈\approx 2.5497854692.549785469
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

−5.11171818288867438888292609926, −4.78269923429936598028536343139, −4.64982929997061365432111977485, −4.63183174145057538038395255810, −4.36223527195295732273681589980, −4.10730708062395728568942631491, −3.94731023187876966275697102425, −3.91008759763318696290211277627, −3.87403364941065329325446591857, −3.25242509453202029788493913655, −3.23531791428257812801357020721, −3.05450190799556185236435560736, −2.92970900312235734574531812403, −2.68119735105710948267816707283, −2.64252810369303682756033193900, −2.62788764105150553599212179629, −2.32965637623613612762920136390, −2.22509669275629148811609855287, −2.02484172789241170997944818127, −1.61383289075840042996961143375, −1.23068546542277125054153530126, −1.02576401867867564325747385002, −0.73764770233626992639436083242, −0.67100685482534905698289186302, −0.25651970391370069379619951743, 0.25651970391370069379619951743, 0.67100685482534905698289186302, 0.73764770233626992639436083242, 1.02576401867867564325747385002, 1.23068546542277125054153530126, 1.61383289075840042996961143375, 2.02484172789241170997944818127, 2.22509669275629148811609855287, 2.32965637623613612762920136390, 2.62788764105150553599212179629, 2.64252810369303682756033193900, 2.68119735105710948267816707283, 2.92970900312235734574531812403, 3.05450190799556185236435560736, 3.23531791428257812801357020721, 3.25242509453202029788493913655, 3.87403364941065329325446591857, 3.91008759763318696290211277627, 3.94731023187876966275697102425, 4.10730708062395728568942631491, 4.36223527195295732273681589980, 4.63183174145057538038395255810, 4.64982929997061365432111977485, 4.78269923429936598028536343139, 5.11171818288867438888292609926

Graph of the ZZ-function along the critical line

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