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Properties

Label 12-4840e6-1.1-c1e6-0-4
Degree 1212
Conductor 1.286×10221.286\times 10^{22}
Sign 11
Analytic cond. 3.33222×1093.33222\times 10^{9}
Root an. cond. 6.216716.21671
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·5-s − 4·7-s − 5·9-s − 12·15-s − 8·17-s − 12·19-s + 8·21-s − 8·23-s + 21·25-s + 10·27-s − 16·29-s − 4·31-s − 24·35-s + 8·37-s − 32·41-s + 4·43-s − 30·45-s − 6·47-s − 5·49-s + 16·51-s + 8·53-s + 24·57-s + 4·59-s − 16·61-s + 20·63-s − 2·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.68·5-s − 1.51·7-s − 5/3·9-s − 3.09·15-s − 1.94·17-s − 2.75·19-s + 1.74·21-s − 1.66·23-s + 21/5·25-s + 1.92·27-s − 2.97·29-s − 0.718·31-s − 4.05·35-s + 1.31·37-s − 4.99·41-s + 0.609·43-s − 4.47·45-s − 0.875·47-s − 5/7·49-s + 2.24·51-s + 1.09·53-s + 3.17·57-s + 0.520·59-s − 2.04·61-s + 2.51·63-s − 0.244·67-s + ⋯

Functional equation

Λ(s)=((218⋅56⋅1112)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅56⋅1112)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅56⋅11122^{18} \cdot 5^{6} \cdot 11^{12}
Sign: 11
Analytic conductor: 3.33222×1093.33222\times 10^{9}
Root analytic conductor: 6.216716.21671
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 218⋅56⋅1112, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 (1−T)6 ( 1 - T )^{6}
11 1 1
good3 1+2T+p2T2+2p2T3+50T4+82T5+181T6+82pT7+50p2T8+2p5T9+p6T10+2p5T11+p6T12 1 + 2 T + p^{2} T^{2} + 2 p^{2} T^{3} + 50 T^{4} + 82 T^{5} + 181 T^{6} + 82 p T^{7} + 50 p^{2} T^{8} + 2 p^{5} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
7 1+4T+3pT2+8pT3+218T4+636T5+1993T6+636pT7+218p2T8+8p4T9+3p5T10+4p5T11+p6T12 1 + 4 T + 3 p T^{2} + 8 p T^{3} + 218 T^{4} + 636 T^{5} + 1993 T^{6} + 636 p T^{7} + 218 p^{2} T^{8} + 8 p^{4} T^{9} + 3 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
13 1+46T2+16T3+935T4+560T5+13172T6+560pT7+935p2T8+16p3T9+46p4T10+p6T12 1 + 46 T^{2} + 16 T^{3} + 935 T^{4} + 560 T^{5} + 13172 T^{6} + 560 p T^{7} + 935 p^{2} T^{8} + 16 p^{3} T^{9} + 46 p^{4} T^{10} + p^{6} T^{12}
17 1+8T+86T2+488T3+191pT4+14352T5+70772T6+14352pT7+191p3T8+488p3T9+86p4T10+8p5T11+p6T12 1 + 8 T + 86 T^{2} + 488 T^{3} + 191 p T^{4} + 14352 T^{5} + 70772 T^{6} + 14352 p T^{7} + 191 p^{3} T^{8} + 488 p^{3} T^{9} + 86 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
19 1+12T+142T2+1028T3+6983T4+36232T5+176372T6+36232pT7+6983p2T8+1028p3T9+142p4T10+12p5T11+p6T12 1 + 12 T + 142 T^{2} + 1028 T^{3} + 6983 T^{4} + 36232 T^{5} + 176372 T^{6} + 36232 p T^{7} + 6983 p^{2} T^{8} + 1028 p^{3} T^{9} + 142 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
23 1+8T+78T2+464T3+3647T4+17448T5+99796T6+17448pT7+3647p2T8+464p3T9+78p4T10+8p5T11+p6T12 1 + 8 T + 78 T^{2} + 464 T^{3} + 3647 T^{4} + 17448 T^{5} + 99796 T^{6} + 17448 p T^{7} + 3647 p^{2} T^{8} + 464 p^{3} T^{9} + 78 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
29 1+16T+158T2+976T3+3655T4+5856T5−10876T6+5856pT7+3655p2T8+976p3T9+158p4T10+16p5T11+p6T12 1 + 16 T + 158 T^{2} + 976 T^{3} + 3655 T^{4} + 5856 T^{5} - 10876 T^{6} + 5856 p T^{7} + 3655 p^{2} T^{8} + 976 p^{3} T^{9} + 158 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
31 1+4T+106T2+580T3+5471T4+35008T5+194684T6+35008pT7+5471p2T8+580p3T9+106p4T10+4p5T11+p6T12 1 + 4 T + 106 T^{2} + 580 T^{3} + 5471 T^{4} + 35008 T^{5} + 194684 T^{6} + 35008 p T^{7} + 5471 p^{2} T^{8} + 580 p^{3} T^{9} + 106 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
37 1−8T+142T2−872T3+9911T4−52016T5+449252T6−52016pT7+9911p2T8−872p3T9+142p4T10−8p5T11+p6T12 1 - 8 T + 142 T^{2} - 872 T^{3} + 9911 T^{4} - 52016 T^{5} + 449252 T^{6} - 52016 p T^{7} + 9911 p^{2} T^{8} - 872 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
41 1+32T+597T2+7760T3+78494T4+647040T5+4497049T6+647040pT7+78494p2T8+7760p3T9+597p4T10+32p5T11+p6T12 1 + 32 T + 597 T^{2} + 7760 T^{3} + 78494 T^{4} + 647040 T^{5} + 4497049 T^{6} + 647040 p T^{7} + 78494 p^{2} T^{8} + 7760 p^{3} T^{9} + 597 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12}
43 1−4T+157T2−160T3+9938T4+11060T5+442193T6+11060pT7+9938p2T8−160p3T9+157p4T10−4p5T11+p6T12 1 - 4 T + 157 T^{2} - 160 T^{3} + 9938 T^{4} + 11060 T^{5} + 442193 T^{6} + 11060 p T^{7} + 9938 p^{2} T^{8} - 160 p^{3} T^{9} + 157 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
47 1+6T+217T2+1142T3+22146T4+96502T5+1330941T6+96502pT7+22146p2T8+1142p3T9+217p4T10+6p5T11+p6T12 1 + 6 T + 217 T^{2} + 1142 T^{3} + 22146 T^{4} + 96502 T^{5} + 1330941 T^{6} + 96502 p T^{7} + 22146 p^{2} T^{8} + 1142 p^{3} T^{9} + 217 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
53 1−8T+142T2−624T3+8055T4−33704T5+443124T6−33704pT7+8055p2T8−624p3T9+142p4T10−8p5T11+p6T12 1 - 8 T + 142 T^{2} - 624 T^{3} + 8055 T^{4} - 33704 T^{5} + 443124 T^{6} - 33704 p T^{7} + 8055 p^{2} T^{8} - 624 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
59 1−4T+106T2−804T3+11415T4−59440T5+798588T6−59440pT7+11415p2T8−804p3T9+106p4T10−4p5T11+p6T12 1 - 4 T + 106 T^{2} - 804 T^{3} + 11415 T^{4} - 59440 T^{5} + 798588 T^{6} - 59440 p T^{7} + 11415 p^{2} T^{8} - 804 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
61 1+16T+245T2+2240T3+24374T4+199632T5+1856313T6+199632pT7+24374p2T8+2240p3T9+245p4T10+16p5T11+p6T12 1 + 16 T + 245 T^{2} + 2240 T^{3} + 24374 T^{4} + 199632 T^{5} + 1856313 T^{6} + 199632 p T^{7} + 24374 p^{2} T^{8} + 2240 p^{3} T^{9} + 245 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
67 1+2T+49T2+458T3+11018T4+49970T5+322277T6+49970pT7+11018p2T8+458p3T9+49p4T10+2p5T11+p6T12 1 + 2 T + 49 T^{2} + 458 T^{3} + 11018 T^{4} + 49970 T^{5} + 322277 T^{6} + 49970 p T^{7} + 11018 p^{2} T^{8} + 458 p^{3} T^{9} + 49 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
71 1+28T+546T2+7900T3+98591T4+1012032T5+9200236T6+1012032pT7+98591p2T8+7900p3T9+546p4T10+28p5T11+p6T12 1 + 28 T + 546 T^{2} + 7900 T^{3} + 98591 T^{4} + 1012032 T^{5} + 9200236 T^{6} + 1012032 p T^{7} + 98591 p^{2} T^{8} + 7900 p^{3} T^{9} + 546 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12}
73 1+16T+246T2+2768T3+28991T4+286368T5+2601844T6+286368pT7+28991p2T8+2768p3T9+246p4T10+16p5T11+p6T12 1 + 16 T + 246 T^{2} + 2768 T^{3} + 28991 T^{4} + 286368 T^{5} + 2601844 T^{6} + 286368 p T^{7} + 28991 p^{2} T^{8} + 2768 p^{3} T^{9} + 246 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
79 1+342T2+288T3+55791T4+50976T5+5551220T6+50976pT7+55791p2T8+288p3T9+342p4T10+p6T12 1 + 342 T^{2} + 288 T^{3} + 55791 T^{4} + 50976 T^{5} + 5551220 T^{6} + 50976 p T^{7} + 55791 p^{2} T^{8} + 288 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12}
83 1+12T+210T2+1028T3+22311T4+168072T5+2881772T6+168072pT7+22311p2T8+1028p3T9+210p4T10+12p5T11+p6T12 1 + 12 T + 210 T^{2} + 1028 T^{3} + 22311 T^{4} + 168072 T^{5} + 2881772 T^{6} + 168072 p T^{7} + 22311 p^{2} T^{8} + 1028 p^{3} T^{9} + 210 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
89 1−18T+253T2−1798T3+8058T4+72934T5−1088091T6+72934pT7+8058p2T8−1798p3T9+253p4T10−18p5T11+p6T12 1 - 18 T + 253 T^{2} - 1798 T^{3} + 8058 T^{4} + 72934 T^{5} - 1088091 T^{6} + 72934 p T^{7} + 8058 p^{2} T^{8} - 1798 p^{3} T^{9} + 253 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
97 1+342T2+1080T3+57231T4+248616T5+6595652T6+248616pT7+57231p2T8+1080p3T9+342p4T10+p6T12 1 + 342 T^{2} + 1080 T^{3} + 57231 T^{4} + 248616 T^{5} + 6595652 T^{6} + 248616 p T^{7} + 57231 p^{2} T^{8} + 1080 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12}
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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

−4.82940116053496188437579965137, −4.38697346658244414764206754907, −4.35028037059269154736930087422, −4.19144598231374951704796550641, −4.17402206086602341137051794799, −4.06553203345657894910602162357, −3.99280257085995928080592881090, −3.52062240393997614157804764761, −3.37222660974894868823018848867, −3.32921233561117657834995531154, −3.21061711648312574027150669985, −3.16405449445578732334569760020, −3.10575758315605535131095567112, −2.59117423742805720460575045348, −2.57125622261674315951456214531, −2.45450545892824961281589036281, −2.17253712569091146842021731432, −2.16757284309816279939303211315, −2.15175402636610473321615796557, −1.88672802065170928107240877298, −1.71710013420403844786507845850, −1.50195126474200778674830870961, −1.44676036649645738241952603334, −1.12300771710951077959255575604, −1.01155350655029705685685737547, 0, 0, 0, 0, 0, 0, 1.01155350655029705685685737547, 1.12300771710951077959255575604, 1.44676036649645738241952603334, 1.50195126474200778674830870961, 1.71710013420403844786507845850, 1.88672802065170928107240877298, 2.15175402636610473321615796557, 2.16757284309816279939303211315, 2.17253712569091146842021731432, 2.45450545892824961281589036281, 2.57125622261674315951456214531, 2.59117423742805720460575045348, 3.10575758315605535131095567112, 3.16405449445578732334569760020, 3.21061711648312574027150669985, 3.32921233561117657834995531154, 3.37222660974894868823018848867, 3.52062240393997614157804764761, 3.99280257085995928080592881090, 4.06553203345657894910602162357, 4.17402206086602341137051794799, 4.19144598231374951704796550641, 4.35028037059269154736930087422, 4.38697346658244414764206754907, 4.82940116053496188437579965137

Graph of the ZZ-function along the critical line

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