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Properties

Label 12-3800e6-1.1-c1e6-0-9
Degree 1212
Conductor 3.011×10213.011\times 10^{21}
Sign 11
Analytic cond. 7.80484×1087.80484\times 10^{8}
Root an. cond. 5.508465.50846
Motivic weight 11
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 − 2·7-s − 2·9-s + 3·11-s + 3·13-s + 2·17-s + 6·19-s + 4·21-s − 4·23-s + 2·27-s + 7·29-s + 5·31-s − 6·33-s − 6·39-s + 11·41-s + 7·43-s − 20·47-s − 20·49-s − 4·51-s + 7·53-s − 12·57-s − 4·59-s + 13·61-s + 4·63-s − 25·67-s + 8·69-s + 29·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s + 0.485·17-s + 1.37·19-s + 0.872·21-s − 0.834·23-s + 0.384·27-s + 1.29·29-s + 0.898·31-s − 1.04·33-s − 0.960·39-s + 1.71·41-s + 1.06·43-s − 2.91·47-s − 2.85·49-s − 0.560·51-s + 0.961·53-s − 1.58·57-s − 0.520·59-s + 1.66·61-s + 0.503·63-s − 3.05·67-s + 0.963·69-s + 3.44·71-s + ⋯

Functional equation

Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅512⋅196)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅512⋅1962^{18} \cdot 5^{12} \cdot 19^{6}
Sign: 11
Analytic conductor: 7.80484×1087.80484\times 10^{8}
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅512⋅196, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.6403226006.640322600
L(12)L(\frac12) ≈\approx 6.6403226006.640322600
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 1
19 (1−T)6 ( 1 - T )^{6}
good3 1+2T+2pT2+14T3+8pT4+40T5+83T6+40pT7+8p3T8+14p3T9+2p5T10+2p5T11+p6T12 1 + 2 T + 2 p T^{2} + 14 T^{3} + 8 p T^{4} + 40 T^{5} + 83 T^{6} + 40 p T^{7} + 8 p^{3} T^{8} + 14 p^{3} T^{9} + 2 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
7 1+2T+24T2+54T3+318T4+594T5+2791T6+594pT7+318p2T8+54p3T9+24p4T10+2p5T11+p6T12 1 + 2 T + 24 T^{2} + 54 T^{3} + 318 T^{4} + 594 T^{5} + 2791 T^{6} + 594 p T^{7} + 318 p^{2} T^{8} + 54 p^{3} T^{9} + 24 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
11 1−3T+14T2−50T3+433T4−915T5+3400T6−915pT7+433p2T8−50p3T9+14p4T10−3p5T11+p6T12 1 - 3 T + 14 T^{2} - 50 T^{3} + 433 T^{4} - 915 T^{5} + 3400 T^{6} - 915 p T^{7} + 433 p^{2} T^{8} - 50 p^{3} T^{9} + 14 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
13 1−3T+35T2−134T3+790T4−2388T5+13141T6−2388pT7+790p2T8−134p3T9+35p4T10−3p5T11+p6T12 1 - 3 T + 35 T^{2} - 134 T^{3} + 790 T^{4} - 2388 T^{5} + 13141 T^{6} - 2388 p T^{7} + 790 p^{2} T^{8} - 134 p^{3} T^{9} + 35 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
17 1−2T+40T2+12T3+422T4+2878T5+309T6+2878pT7+422p2T8+12p3T9+40p4T10−2p5T11+p6T12 1 - 2 T + 40 T^{2} + 12 T^{3} + 422 T^{4} + 2878 T^{5} + 309 T^{6} + 2878 p T^{7} + 422 p^{2} T^{8} + 12 p^{3} T^{9} + 40 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
23 1+4T+96T2+346T3+4540T4+14008T5+131019T6+14008pT7+4540p2T8+346p3T9+96p4T10+4p5T11+p6T12 1 + 4 T + 96 T^{2} + 346 T^{3} + 4540 T^{4} + 14008 T^{5} + 131019 T^{6} + 14008 p T^{7} + 4540 p^{2} T^{8} + 346 p^{3} T^{9} + 96 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
29 1−7T+101T2−680T3+6104T4−32894T5+218211T6−32894pT7+6104p2T8−680p3T9+101p4T10−7p5T11+p6T12 1 - 7 T + 101 T^{2} - 680 T^{3} + 6104 T^{4} - 32894 T^{5} + 218211 T^{6} - 32894 p T^{7} + 6104 p^{2} T^{8} - 680 p^{3} T^{9} + 101 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}
31 1−5T+116T2−166T3+4207T4+9815T5+97168T6+9815pT7+4207p2T8−166p3T9+116p4T10−5p5T11+p6T12 1 - 5 T + 116 T^{2} - 166 T^{3} + 4207 T^{4} + 9815 T^{5} + 97168 T^{6} + 9815 p T^{7} + 4207 p^{2} T^{8} - 166 p^{3} T^{9} + 116 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
37 1+14T2−189T3+1675T4−1758T5+75767T6−1758pT7+1675p2T8−189p3T9+14p4T10+p6T12 1 + 14 T^{2} - 189 T^{3} + 1675 T^{4} - 1758 T^{5} + 75767 T^{6} - 1758 p T^{7} + 1675 p^{2} T^{8} - 189 p^{3} T^{9} + 14 p^{4} T^{10} + p^{6} T^{12}
41 1−11T+162T2−1616T3+14575T4−107273T5+784764T6−107273pT7+14575p2T8−1616p3T9+162p4T10−11p5T11+p6T12 1 - 11 T + 162 T^{2} - 1616 T^{3} + 14575 T^{4} - 107273 T^{5} + 784764 T^{6} - 107273 p T^{7} + 14575 p^{2} T^{8} - 1616 p^{3} T^{9} + 162 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12}
43 1−7T+pT2−530T3+3441T4−21767T5+216350T6−21767pT7+3441p2T8−530p3T9+p5T10−7p5T11+p6T12 1 - 7 T + p T^{2} - 530 T^{3} + 3441 T^{4} - 21767 T^{5} + 216350 T^{6} - 21767 p T^{7} + 3441 p^{2} T^{8} - 530 p^{3} T^{9} + p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}
47 1+20T+294T2+2363T3+14371T4+39728T5+169611T6+39728pT7+14371p2T8+2363p3T9+294p4T10+20p5T11+p6T12 1 + 20 T + 294 T^{2} + 2363 T^{3} + 14371 T^{4} + 39728 T^{5} + 169611 T^{6} + 39728 p T^{7} + 14371 p^{2} T^{8} + 2363 p^{3} T^{9} + 294 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12}
53 1−7T+286T2−1674T3+35709T4−168565T5+2474973T6−168565pT7+35709p2T8−1674p3T9+286p4T10−7p5T11+p6T12 1 - 7 T + 286 T^{2} - 1674 T^{3} + 35709 T^{4} - 168565 T^{5} + 2474973 T^{6} - 168565 p T^{7} + 35709 p^{2} T^{8} - 1674 p^{3} T^{9} + 286 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}
59 1+4T+261T2+825T3+32193T4+80901T5+2380506T6+80901pT7+32193p2T8+825p3T9+261p4T10+4p5T11+p6T12 1 + 4 T + 261 T^{2} + 825 T^{3} + 32193 T^{4} + 80901 T^{5} + 2380506 T^{6} + 80901 p T^{7} + 32193 p^{2} T^{8} + 825 p^{3} T^{9} + 261 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
61 1−13T+246T2−2418T3+29973T4−235869T5+2276056T6−235869pT7+29973p2T8−2418p3T9+246p4T10−13p5T11+p6T12 1 - 13 T + 246 T^{2} - 2418 T^{3} + 29973 T^{4} - 235869 T^{5} + 2276056 T^{6} - 235869 p T^{7} + 29973 p^{2} T^{8} - 2418 p^{3} T^{9} + 246 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12}
67 1+25T+505T2+6772T3+80896T4+773578T5+6921907T6+773578pT7+80896p2T8+6772p3T9+505p4T10+25p5T11+p6T12 1 + 25 T + 505 T^{2} + 6772 T^{3} + 80896 T^{4} + 773578 T^{5} + 6921907 T^{6} + 773578 p T^{7} + 80896 p^{2} T^{8} + 6772 p^{3} T^{9} + 505 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12}
71 1−29T+603T2−8526T3+102909T4−1014029T5+9209494T6−1014029pT7+102909p2T8−8526p3T9+603p4T10−29p5T11+p6T12 1 - 29 T + 603 T^{2} - 8526 T^{3} + 102909 T^{4} - 1014029 T^{5} + 9209494 T^{6} - 1014029 p T^{7} + 102909 p^{2} T^{8} - 8526 p^{3} T^{9} + 603 p^{4} T^{10} - 29 p^{5} T^{11} + p^{6} T^{12}
73 1−19T+505T2−6558T3+98292T4−932744T5+9748099T6−932744pT7+98292p2T8−6558p3T9+505p4T10−19p5T11+p6T12 1 - 19 T + 505 T^{2} - 6558 T^{3} + 98292 T^{4} - 932744 T^{5} + 9748099 T^{6} - 932744 p T^{7} + 98292 p^{2} T^{8} - 6558 p^{3} T^{9} + 505 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12}
79 1−28T+659T2−10371T3+145815T4−1597403T5+15814250T6−1597403pT7+145815p2T8−10371p3T9+659p4T10−28p5T11+p6T12 1 - 28 T + 659 T^{2} - 10371 T^{3} + 145815 T^{4} - 1597403 T^{5} + 15814250 T^{6} - 1597403 p T^{7} + 145815 p^{2} T^{8} - 10371 p^{3} T^{9} + 659 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12}
83 1−15T+496T2−5598T3+100571T4−875935T5+11004272T6−875935pT7+100571p2T8−5598p3T9+496p4T10−15p5T11+p6T12 1 - 15 T + 496 T^{2} - 5598 T^{3} + 100571 T^{4} - 875935 T^{5} + 11004272 T^{6} - 875935 p T^{7} + 100571 p^{2} T^{8} - 5598 p^{3} T^{9} + 496 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12}
89 1+12T+317T2+3161T3+56077T4+457091T5+6094922T6+457091pT7+56077p2T8+3161p3T9+317p4T10+12p5T11+p6T12 1 + 12 T + 317 T^{2} + 3161 T^{3} + 56077 T^{4} + 457091 T^{5} + 6094922 T^{6} + 457091 p T^{7} + 56077 p^{2} T^{8} + 3161 p^{3} T^{9} + 317 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
97 1−13T+448T2−3788T3+80619T4−494279T5+9036752T6−494279pT7+80619p2T8−3788p3T9+448p4T10−13p5T11+p6T12 1 - 13 T + 448 T^{2} - 3788 T^{3} + 80619 T^{4} - 494279 T^{5} + 9036752 T^{6} - 494279 p T^{7} + 80619 p^{2} T^{8} - 3788 p^{3} T^{9} + 448 p^{4} T^{10} - 13 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

−4.46648786347518582615362076038, −4.24279194898142944047893608085, −3.93283401751500165448886812878, −3.85816143555688724288215696468, −3.80934785976132510067552783818, −3.69035037313820857998537092097, −3.62594136921848654234886457068, −3.34852421856841495755656570253, −3.07814905223241342159665174421, −3.00755292430544107951787788499, −2.98611152398659657223707538927, −2.89417324206712532907744656805, −2.86946963991985976446731019380, −2.20796277163223607968511295493, −2.11319549206465084375759513915, −2.09090628458287860144196322609, −2.01941328226326698882869417820, −1.74476405464845389362770201299, −1.58114325876011649506517240097, −1.19296284502594746923767920620, −1.01419590745058069875263520790, −0.818379065893886754149673685414, −0.63634959477184138414104289188, −0.47486822745821865085859932953, −0.39984940269767498431159631899, 0.39984940269767498431159631899, 0.47486822745821865085859932953, 0.63634959477184138414104289188, 0.818379065893886754149673685414, 1.01419590745058069875263520790, 1.19296284502594746923767920620, 1.58114325876011649506517240097, 1.74476405464845389362770201299, 2.01941328226326698882869417820, 2.09090628458287860144196322609, 2.11319549206465084375759513915, 2.20796277163223607968511295493, 2.86946963991985976446731019380, 2.89417324206712532907744656805, 2.98611152398659657223707538927, 3.00755292430544107951787788499, 3.07814905223241342159665174421, 3.34852421856841495755656570253, 3.62594136921848654234886457068, 3.69035037313820857998537092097, 3.80934785976132510067552783818, 3.85816143555688724288215696468, 3.93283401751500165448886812878, 4.24279194898142944047893608085, 4.46648786347518582615362076038

Graph of the ZZ-function along the critical line

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