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Properties

Label 12-1815e6-1.1-c3e6-0-1
Degree 1212
Conductor 3.575×10193.575\times 10^{19}
Sign 11
Analytic cond. 1.50819×10121.50819\times 10^{12}
Root an. cond. 10.348310.3483
Motivic weight 33
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  + 18·3-s − 4·4-s + 30·5-s + 189·9-s − 72·12-s + 540·15-s − 31·16-s − 120·20-s + 56·23-s + 525·25-s + 1.51e3·27-s + 576·31-s − 756·36-s + 332·37-s + 5.67e3·45-s + 96·47-s − 558·48-s − 802·49-s + 308·53-s + 2.08e3·59-s − 2.16e3·60-s + 248·64-s − 1.16e3·67-s + 1.00e3·69-s + 1.06e3·71-s + 9.45e3·75-s − 930·80-s + ⋯
L(s)  = 1  + 3.46·3-s − 1/2·4-s + 2.68·5-s + 7·9-s − 1.73·12-s + 9.29·15-s − 0.484·16-s − 1.34·20-s + 0.507·23-s + 21/5·25-s + 10.7·27-s + 3.33·31-s − 7/2·36-s + 1.47·37-s + 18.7·45-s + 0.297·47-s − 1.67·48-s − 2.33·49-s + 0.798·53-s + 4.58·59-s − 4.64·60-s + 0.484·64-s − 2.12·67-s + 1.75·69-s + 1.77·71-s + 14.5·75-s − 1.29·80-s + ⋯

Functional equation

Λ(s)=((36⋅56⋅1112)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((36⋅56⋅1112)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 36⋅56⋅11123^{6} \cdot 5^{6} \cdot 11^{12}
Sign: 11
Analytic conductor: 1.50819×10121.50819\times 10^{12}
Root analytic conductor: 10.348310.3483
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 36⋅56⋅1112, ( :[3/2]6), 1)(12,\ 3^{6} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 200.1380594200.1380594
L(12)L(\frac12) ≈\approx 200.1380594200.1380594
L(52)L(\frac{5}{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)
bad3 (1−pT)6 ( 1 - p T )^{6}
5 (1−pT)6 ( 1 - p T )^{6}
11 1 1
good2 1+p2T2+47T4+p6T6+47p6T8+p14T10+p18T12 1 + p^{2} T^{2} + 47 T^{4} + p^{6} T^{6} + 47 p^{6} T^{8} + p^{14} T^{10} + p^{18} T^{12}
7 1+802T2+304543T4+94343516T6+304543p6T8+802p12T10+p18T12 1 + 802 T^{2} + 304543 T^{4} + 94343516 T^{6} + 304543 p^{6} T^{8} + 802 p^{12} T^{10} + p^{18} T^{12}
13 1+886pT2+58019383T4+164950568324T6+58019383p6T8+886p13T10+p18T12 1 + 886 p T^{2} + 58019383 T^{4} + 164950568324 T^{6} + 58019383 p^{6} T^{8} + 886 p^{13} T^{10} + p^{18} T^{12}
17 1−1346T2−18224113T4+91405101604T6−18224113p6T8−1346p12T10+p18T12 1 - 1346 T^{2} - 18224113 T^{4} + 91405101604 T^{6} - 18224113 p^{6} T^{8} - 1346 p^{12} T^{10} + p^{18} T^{12}
19 1+1462pT2+381628039T4+3260386698524T6+381628039p6T8+1462p13T10+p18T12 1 + 1462 p T^{2} + 381628039 T^{4} + 3260386698524 T^{6} + 381628039 p^{6} T^{8} + 1462 p^{13} T^{10} + p^{18} T^{12}
23 (1−28T+9957T2+1257080T3+9957p3T4−28p6T5+p9T6)2 ( 1 - 28 T + 9957 T^{2} + 1257080 T^{3} + 9957 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} )^{2}
29 1+107278T2+5613031079T4+173083909993924T6+5613031079p6T8+107278p12T10+p18T12 1 + 107278 T^{2} + 5613031079 T^{4} + 173083909993924 T^{6} + 5613031079 p^{6} T^{8} + 107278 p^{12} T^{10} + p^{18} T^{12}
31 (1−288T+95373T2−16398016T3+95373p3T4−288p6T5+p9T6)2 ( 1 - 288 T + 95373 T^{2} - 16398016 T^{3} + 95373 p^{3} T^{4} - 288 p^{6} T^{5} + p^{9} T^{6} )^{2}
37 (1−166T+134339T2−13813924T3+134339p3T4−166p6T5+p9T6)2 ( 1 - 166 T + 134339 T^{2} - 13813924 T^{3} + 134339 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} )^{2}
41 1+392550T2+65515823631T4+5956276963486324T6+65515823631p6T8+392550p12T10+p18T12 1 + 392550 T^{2} + 65515823631 T^{4} + 5956276963486324 T^{6} + 65515823631 p^{6} T^{8} + 392550 p^{12} T^{10} + p^{18} T^{12}
43 1+167898T2+17757275463T4+1598829373108364T6+17757275463p6T8+167898p12T10+p18T12 1 + 167898 T^{2} + 17757275463 T^{4} + 1598829373108364 T^{6} + 17757275463 p^{6} T^{8} + 167898 p^{12} T^{10} + p^{18} T^{12}
47 (1−48T+187437T2+3225696T3+187437p3T4−48p6T5+p9T6)2 ( 1 - 48 T + 187437 T^{2} + 3225696 T^{3} + 187437 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} )^{2}
53 (1−154T+250803T2−59517148T3+250803p3T4−154p6T5+p9T6)2 ( 1 - 154 T + 250803 T^{2} - 59517148 T^{3} + 250803 p^{3} T^{4} - 154 p^{6} T^{5} + p^{9} T^{6} )^{2}
59 (1−1040T+659337T2−334284320T3+659337p3T4−1040p6T5+p9T6)2 ( 1 - 1040 T + 659337 T^{2} - 334284320 T^{3} + 659337 p^{3} T^{4} - 1040 p^{6} T^{5} + p^{9} T^{6} )^{2}
61 1+1115710T2+559401461431T4+162322226576222084T6+559401461431p6T8+1115710p12T10+p18T12 1 + 1115710 T^{2} + 559401461431 T^{4} + 162322226576222084 T^{6} + 559401461431 p^{6} T^{8} + 1115710 p^{12} T^{10} + p^{18} T^{12}
67 (1+584T+19793T2−187697680T3+19793p3T4+584p6T5+p9T6)2 ( 1 + 584 T + 19793 T^{2} - 187697680 T^{3} + 19793 p^{3} T^{4} + 584 p^{6} T^{5} + p^{9} T^{6} )^{2}
71 (1−532T+371829T2−482157976T3+371829p3T4−532p6T5+p9T6)2 ( 1 - 532 T + 371829 T^{2} - 482157976 T^{3} + 371829 p^{3} T^{4} - 532 p^{6} T^{5} + p^{9} T^{6} )^{2}
73 1+1476678T2+1108133530623T4+524326150985219924T6+1108133530623p6T8+1476678p12T10+p18T12 1 + 1476678 T^{2} + 1108133530623 T^{4} + 524326150985219924 T^{6} + 1108133530623 p^{6} T^{8} + 1476678 p^{12} T^{10} + p^{18} T^{12}
79 1+649210T2+356801976031T4+97641164525431436T6+356801976031p6T8+649210p12T10+p18T12 1 + 649210 T^{2} + 356801976031 T^{4} + 97641164525431436 T^{6} + 356801976031 p^{6} T^{8} + 649210 p^{12} T^{10} + p^{18} T^{12}
83 1+2530162T2+2916380468855T4+2053315560734357020T6+2916380468855p6T8+2530162p12T10+p18T12 1 + 2530162 T^{2} + 2916380468855 T^{4} + 2053315560734357020 T^{6} + 2916380468855 p^{6} T^{8} + 2530162 p^{12} T^{10} + p^{18} T^{12}
89 (1+342T+2024103T2+471201876T3+2024103p3T4+342p6T5+p9T6)2 ( 1 + 342 T + 2024103 T^{2} + 471201876 T^{3} + 2024103 p^{3} T^{4} + 342 p^{6} T^{5} + p^{9} T^{6} )^{2}
97 (1−1406T+1951919T2−1639433444T3+1951919p3T4−1406p6T5+p9T6)2 ( 1 - 1406 T + 1951919 T^{2} - 1639433444 T^{3} + 1951919 p^{3} T^{4} - 1406 p^{6} T^{5} + p^{9} T^{6} )^{2}
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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

−4.65373309934304937388015097981, −4.06578856020299078175997031077, −3.97379743618628166860216641732, −3.93732599849672536407076968952, −3.87414121501668631345885518995, −3.75033995049429060257948107802, −3.41739047889294735133890554364, −3.17383396434966958745570282928, −3.09247010436275058565313815379, −2.87830680894750883760537362042, −2.80037734563663105944052343802, −2.55413511531942314166241788734, −2.54611547018043129420590280641, −2.51132424948967309616672570234, −2.19244778740010749458144788440, −2.06026352859916346685841394232, −1.90064964020114025610781533867, −1.61923964591889839547319110050, −1.61779266309966773239668010028, −1.25566548165062588526259291134, −1.18083421874254433486135847690, −0.844883549829202385190923874806, −0.75158168061831881524838967917, −0.64395549302188950099032247497, −0.27604139538996374723181536907, 0.27604139538996374723181536907, 0.64395549302188950099032247497, 0.75158168061831881524838967917, 0.844883549829202385190923874806, 1.18083421874254433486135847690, 1.25566548165062588526259291134, 1.61779266309966773239668010028, 1.61923964591889839547319110050, 1.90064964020114025610781533867, 2.06026352859916346685841394232, 2.19244778740010749458144788440, 2.51132424948967309616672570234, 2.54611547018043129420590280641, 2.55413511531942314166241788734, 2.80037734563663105944052343802, 2.87830680894750883760537362042, 3.09247010436275058565313815379, 3.17383396434966958745570282928, 3.41739047889294735133890554364, 3.75033995049429060257948107802, 3.87414121501668631345885518995, 3.93732599849672536407076968952, 3.97379743618628166860216641732, 4.06578856020299078175997031077, 4.65373309934304937388015097981

Graph of the ZZ-function along the critical line

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