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Properties

Label 12-3808e6-1.1-c1e6-0-8
Degree 1212
Conductor 3.049×10213.049\times 10^{21}
Sign 11
Analytic cond. 7.90395×1087.90395\times 10^{8}
Root an. cond. 5.514255.51425
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 − 6·7-s − 5·9-s + 2·11-s − 4·13-s − 12·15-s − 6·17-s + 10·19-s − 12·21-s − 4·23-s + 5·25-s − 12·27-s − 14·29-s − 8·31-s + 4·33-s + 36·35-s − 4·37-s − 8·39-s − 2·41-s + 8·43-s + 30·45-s + 2·47-s + 21·49-s − 12·51-s − 10·53-s − 12·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.68·5-s − 2.26·7-s − 5/3·9-s + 0.603·11-s − 1.10·13-s − 3.09·15-s − 1.45·17-s + 2.29·19-s − 2.61·21-s − 0.834·23-s + 25-s − 2.30·27-s − 2.59·29-s − 1.43·31-s + 0.696·33-s + 6.08·35-s − 0.657·37-s − 1.28·39-s − 0.312·41-s + 1.21·43-s + 4.47·45-s + 0.291·47-s + 3·49-s − 1.68·51-s − 1.37·53-s − 1.61·55-s + ⋯

Functional equation

Λ(s)=((230⋅76⋅176)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((230⋅76⋅176)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 230⋅76⋅1762^{30} \cdot 7^{6} \cdot 17^{6}
Sign: 11
Analytic conductor: 7.90395×1087.90395\times 10^{8}
Root analytic conductor: 5.514255.51425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 230⋅76⋅176, ( :[1/2]6), 1)(12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [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
7 (1+T)6 ( 1 + T )^{6}
17 (1+T)6 ( 1 + T )^{6}
good3 1−2T+p2T2−16T3+16pT4−68T5+56pT6−68pT7+16p3T8−16p3T9+p6T10−2p5T11+p6T12 1 - 2 T + p^{2} T^{2} - 16 T^{3} + 16 p T^{4} - 68 T^{5} + 56 p T^{6} - 68 p T^{7} + 16 p^{3} T^{8} - 16 p^{3} T^{9} + p^{6} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
5 1+6T+31T2+122T3+16p2T4+1106T5+2688T6+1106pT7+16p4T8+122p3T9+31p4T10+6p5T11+p6T12 1 + 6 T + 31 T^{2} + 122 T^{3} + 16 p^{2} T^{4} + 1106 T^{5} + 2688 T^{6} + 1106 p T^{7} + 16 p^{4} T^{8} + 122 p^{3} T^{9} + 31 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
11 1−2T+32T2−18T3+515T4+40T5+6232T6+40pT7+515p2T8−18p3T9+32p4T10−2p5T11+p6T12 1 - 2 T + 32 T^{2} - 18 T^{3} + 515 T^{4} + 40 T^{5} + 6232 T^{6} + 40 p T^{7} + 515 p^{2} T^{8} - 18 p^{3} T^{9} + 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
13 1+4T+4pT2+116T3+971T4+80pT5+12048T6+80p2T7+971p2T8+116p3T9+4p5T10+4p5T11+p6T12 1 + 4 T + 4 p T^{2} + 116 T^{3} + 971 T^{4} + 80 p T^{5} + 12048 T^{6} + 80 p^{2} T^{7} + 971 p^{2} T^{8} + 116 p^{3} T^{9} + 4 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
19 1−10T+82T2−326T3+1027T4+852T5−6444T6+852pT7+1027p2T8−326p3T9+82p4T10−10p5T11+p6T12 1 - 10 T + 82 T^{2} - 326 T^{3} + 1027 T^{4} + 852 T^{5} - 6444 T^{6} + 852 p T^{7} + 1027 p^{2} T^{8} - 326 p^{3} T^{9} + 82 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12}
23 1+4T+2pT2+148T3+1163T4+2656T5+23980T6+2656pT7+1163p2T8+148p3T9+2p5T10+4p5T11+p6T12 1 + 4 T + 2 p T^{2} + 148 T^{3} + 1163 T^{4} + 2656 T^{5} + 23980 T^{6} + 2656 p T^{7} + 1163 p^{2} T^{8} + 148 p^{3} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
29 1+14T+220T2+1906T3+17179T4+106472T5+675888T6+106472pT7+17179p2T8+1906p3T9+220p4T10+14p5T11+p6T12 1 + 14 T + 220 T^{2} + 1906 T^{3} + 17179 T^{4} + 106472 T^{5} + 675888 T^{6} + 106472 p T^{7} + 17179 p^{2} T^{8} + 1906 p^{3} T^{9} + 220 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}
31 1+8T+89T2+664T3+4656T4+25704T5+169068T6+25704pT7+4656p2T8+664p3T9+89p4T10+8p5T11+p6T12 1 + 8 T + 89 T^{2} + 664 T^{3} + 4656 T^{4} + 25704 T^{5} + 169068 T^{6} + 25704 p T^{7} + 4656 p^{2} T^{8} + 664 p^{3} T^{9} + 89 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
37 1+4T+122T2+500T3+8227T4+28312T5+365892T6+28312pT7+8227p2T8+500p3T9+122p4T10+4p5T11+p6T12 1 + 4 T + 122 T^{2} + 500 T^{3} + 8227 T^{4} + 28312 T^{5} + 365892 T^{6} + 28312 p T^{7} + 8227 p^{2} T^{8} + 500 p^{3} T^{9} + 122 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
41 1+2T+195T2+182T3+16804T4+6552T5+861264T6+6552pT7+16804p2T8+182p3T9+195p4T10+2p5T11+p6T12 1 + 2 T + 195 T^{2} + 182 T^{3} + 16804 T^{4} + 6552 T^{5} + 861264 T^{6} + 6552 p T^{7} + 16804 p^{2} T^{8} + 182 p^{3} T^{9} + 195 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
43 1−8T+187T2−1114T3+16156T4−78342T5+862296T6−78342pT7+16156p2T8−1114p3T9+187p4T10−8p5T11+p6T12 1 - 8 T + 187 T^{2} - 1114 T^{3} + 16156 T^{4} - 78342 T^{5} + 862296 T^{6} - 78342 p T^{7} + 16156 p^{2} T^{8} - 1114 p^{3} T^{9} + 187 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
47 1−2T+192T2−250T3+17923T4−17448T5+1041080T6−17448pT7+17923p2T8−250p3T9+192p4T10−2p5T11+p6T12 1 - 2 T + 192 T^{2} - 250 T^{3} + 17923 T^{4} - 17448 T^{5} + 1041080 T^{6} - 17448 p T^{7} + 17923 p^{2} T^{8} - 250 p^{3} T^{9} + 192 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
53 1+10T+225T2+1956T3+23880T4+174814T5+1566748T6+174814pT7+23880p2T8+1956p3T9+225p4T10+10p5T11+p6T12 1 + 10 T + 225 T^{2} + 1956 T^{3} + 23880 T^{4} + 174814 T^{5} + 1566748 T^{6} + 174814 p T^{7} + 23880 p^{2} T^{8} + 1956 p^{3} T^{9} + 225 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12}
59 1−14T+214T2−2370T3+25991T4−223436T5+1918964T6−223436pT7+25991p2T8−2370p3T9+214p4T10−14p5T11+p6T12 1 - 14 T + 214 T^{2} - 2370 T^{3} + 25991 T^{4} - 223436 T^{5} + 1918964 T^{6} - 223436 p T^{7} + 25991 p^{2} T^{8} - 2370 p^{3} T^{9} + 214 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
61 1+20T+263T2+1898T3+3984T4−82348T5−1046808T6−82348pT7+3984p2T8+1898p3T9+263p4T10+20p5T11+p6T12 1 + 20 T + 263 T^{2} + 1898 T^{3} + 3984 T^{4} - 82348 T^{5} - 1046808 T^{6} - 82348 p T^{7} + 3984 p^{2} T^{8} + 1898 p^{3} T^{9} + 263 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12}
67 1−6T+271T2−1610T3+35028T4−188628T5+2861664T6−188628pT7+35028p2T8−1610p3T9+271p4T10−6p5T11+p6T12 1 - 6 T + 271 T^{2} - 1610 T^{3} + 35028 T^{4} - 188628 T^{5} + 2861664 T^{6} - 188628 p T^{7} + 35028 p^{2} T^{8} - 1610 p^{3} T^{9} + 271 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
71 1+20T+446T2+5884T3+79003T4+765512T5+7422124T6+765512pT7+79003p2T8+5884p3T9+446p4T10+20p5T11+p6T12 1 + 20 T + 446 T^{2} + 5884 T^{3} + 79003 T^{4} + 765512 T^{5} + 7422124 T^{6} + 765512 p T^{7} + 79003 p^{2} T^{8} + 5884 p^{3} T^{9} + 446 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12}
73 1+6T+347T2+1634T3+53720T4+200976T5+4919912T6+200976pT7+53720p2T8+1634p3T9+347p4T10+6p5T11+p6T12 1 + 6 T + 347 T^{2} + 1634 T^{3} + 53720 T^{4} + 200976 T^{5} + 4919912 T^{6} + 200976 p T^{7} + 53720 p^{2} T^{8} + 1634 p^{3} T^{9} + 347 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
79 1+38T+740T2+9566T3+97155T4+876392T5+7764912T6+876392pT7+97155p2T8+9566p3T9+740p4T10+38p5T11+p6T12 1 + 38 T + 740 T^{2} + 9566 T^{3} + 97155 T^{4} + 876392 T^{5} + 7764912 T^{6} + 876392 p T^{7} + 97155 p^{2} T^{8} + 9566 p^{3} T^{9} + 740 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12}
83 1−16T+376T2−5088T3+66491T4−748248T5+7035736T6−748248pT7+66491p2T8−5088p3T9+376p4T10−16p5T11+p6T12 1 - 16 T + 376 T^{2} - 5088 T^{3} + 66491 T^{4} - 748248 T^{5} + 7035736 T^{6} - 748248 p T^{7} + 66491 p^{2} T^{8} - 5088 p^{3} T^{9} + 376 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}
89 1+2T+196T2−1210T3+25163T4−107880T5+3544352T6−107880pT7+25163p2T8−1210p3T9+196p4T10+2p5T11+p6T12 1 + 2 T + 196 T^{2} - 1210 T^{3} + 25163 T^{4} - 107880 T^{5} + 3544352 T^{6} - 107880 p T^{7} + 25163 p^{2} T^{8} - 1210 p^{3} T^{9} + 196 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
97 1+12T+435T2+4682T3+89128T4+812778T5+10912024T6+812778pT7+89128p2T8+4682p3T9+435p4T10+12p5T11+p6T12 1 + 12 T + 435 T^{2} + 4682 T^{3} + 89128 T^{4} + 812778 T^{5} + 10912024 T^{6} + 812778 p T^{7} + 89128 p^{2} T^{8} + 4682 p^{3} T^{9} + 435 p^{4} T^{10} + 12 p^{5} T^{11} + 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

−4.68988468230978140532815495230, −4.48085928532999538008344230528, −4.26164874240943837375038611753, −4.18198948711529641343310973155, −4.14916377796797257388094119950, −4.03955590068256630080634915285, −3.95744246604668837048185629908, −3.76890133138156239608886851705, −3.55275701668609376636345551378, −3.51311772738782730522167282922, −3.38222716255013513359990443942, −3.32156312193299551930375717380, −3.11700656696090303244193741452, −2.92667837783300233222455776100, −2.72074148856322114960099326989, −2.62368430784210112605618436452, −2.57942132105659418605102994758, −2.43064576003601370561302715364, −2.35403189183865894639525150762, −1.88087861536073928105515110741, −1.75881643655682183272083048447, −1.62998570287081830709253050734, −1.28650857694416440978908914032, −1.07078851395833386115171509513, −1.03480855013030031661941402520, 0, 0, 0, 0, 0, 0, 1.03480855013030031661941402520, 1.07078851395833386115171509513, 1.28650857694416440978908914032, 1.62998570287081830709253050734, 1.75881643655682183272083048447, 1.88087861536073928105515110741, 2.35403189183865894639525150762, 2.43064576003601370561302715364, 2.57942132105659418605102994758, 2.62368430784210112605618436452, 2.72074148856322114960099326989, 2.92667837783300233222455776100, 3.11700656696090303244193741452, 3.32156312193299551930375717380, 3.38222716255013513359990443942, 3.51311772738782730522167282922, 3.55275701668609376636345551378, 3.76890133138156239608886851705, 3.95744246604668837048185629908, 4.03955590068256630080634915285, 4.14916377796797257388094119950, 4.18198948711529641343310973155, 4.26164874240943837375038611753, 4.48085928532999538008344230528, 4.68988468230978140532815495230

Graph of the ZZ-function along the critical line

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