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Properties

Label 12-1323e6-1.1-c1e6-0-6
Degree 1212
Conductor 5.362×10185.362\times 10^{18}
Sign 11
Analytic cond. 1.39002×1061.39002\times 10^{6}
Root an. cond. 3.250263.25026
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  − 6·2-s + 15·4-s + 3·5-s − 18·8-s − 18·10-s + 6·11-s + 3·13-s + 3·16-s + 6·17-s + 3·19-s + 45·20-s − 36·22-s + 12·23-s + 15·25-s − 18·26-s + 9·29-s − 6·31-s + 30·32-s − 36·34-s + 3·37-s − 18·38-s − 54·40-s + 3·43-s + 90·44-s − 72·46-s − 6·47-s − 90·50-s + ⋯
L(s)  = 1  − 4.24·2-s + 15/2·4-s + 1.34·5-s − 6.36·8-s − 5.69·10-s + 1.80·11-s + 0.832·13-s + 3/4·16-s + 1.45·17-s + 0.688·19-s + 10.0·20-s − 7.67·22-s + 2.50·23-s + 3·25-s − 3.53·26-s + 1.67·29-s − 1.07·31-s + 5.30·32-s − 6.17·34-s + 0.493·37-s − 2.91·38-s − 8.53·40-s + 0.457·43-s + 13.5·44-s − 10.6·46-s − 0.875·47-s − 12.7·50-s + ⋯

Functional equation

Λ(s)=((318⋅712)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((318⋅712)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 318⋅7123^{18} \cdot 7^{12}
Sign: 11
Analytic conductor: 1.39002×1061.39002\times 10^{6}
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 318⋅712, ( :[1/2]6), 1)(12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.68599375690.6859937569
L(12)L(\frac12) ≈\approx 0.68599375690.6859937569
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)
bad3 1 1
7 1 1
good2 (1+3T+3pT2+9T3+3p2T4+3p2T5+p3T6)2 ( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
5 1−3T−6T2+9T3+69T4−6pT5−371T6−6p2T7+69p2T8+9p3T9−6p4T10−3p5T11+p6T12 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
11 1−6T−6T2+18T3+492T4−852T5−2873T6−852pT7+492p2T8+18p3T9−6p4T10−6p5T11+p6T12 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
13 1−3T+3T2−76T3+45T4+135T5+3246T6+135pT7+45p2T8−76p3T9+3p4T10−3p5T11+p6T12 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
17 1−6T−24T2+54T3+1338T4−1914T5−18929T6−1914pT7+1338p2T8+54p3T9−24p4T10−6p5T11+p6T12 1 - 6 T - 24 T^{2} + 54 T^{3} + 1338 T^{4} - 1914 T^{5} - 18929 T^{6} - 1914 p T^{7} + 1338 p^{2} T^{8} + 54 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
19 1−3T−42T2+41T3+1341T4−216T5−29541T6−216pT7+1341p2T8+41p3T9−42p4T10−3p5T11+p6T12 1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
23 1−12T+48T2−54T3+420T4−6060T5+37591T6−6060pT7+420p2T8−54p3T9+48p4T10−12p5T11+p6T12 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
29 1−9T+30T2−81T3−579T4+9414T5−59051T6+9414pT7−579p2T8−81p3T9+30p4T10−9p5T11+p6T12 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
31 (1+3T+15T2−137T3+15pT4+3p2T5+p3T6)2 ( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1−3T−24T2−301T3+171T4+6552T5+58893T6+6552pT7+171p2T8−301p3T9−24p4T10−3p5T11+p6T12 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
41 1−114T2+18T3+8322T4−1026T5−394913T6−1026pT7+8322p2T8+18p3T9−114p4T10+p6T12 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12}
43 1−3T−114T2+149T3+9063T4−5670T5−441093T6−5670pT7+9063p2T8+149p3T9−114p4T10−3p5T11+p6T12 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
47 (1+3T+87T2+333T3+87pT4+3p2T5+p3T6)2 ( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
53 1−6T−114T2+378T3+10716T4−17304T5−587549T6−17304pT7+10716p2T8+378p3T9−114p4T10−6p5T11+p6T12 1 - 6 T - 114 T^{2} + 378 T^{3} + 10716 T^{4} - 17304 T^{5} - 587549 T^{6} - 17304 p T^{7} + 10716 p^{2} T^{8} + 378 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
59 (1−3T+105T2−405T3+105pT4−3p2T5+p3T6)2 ( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1−6T+168T2−713T3+168pT4−6p2T5+p3T6)2 ( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 (1+12T+222T2+1591T3+222pT4+12p2T5+p3T6)2 ( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
71 (1+9T+159T2+1305T3+159pT4+9p2T5+p3T6)2 ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−21T+138T2−769T3+10953T4−30402T5−450903T6−30402pT7+10953p2T8−769p3T9+138p4T10−21p5T11+p6T12 1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12}
79 (1+21T+357T2+3499T3+357pT4+21p2T5+p3T6)2 ( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1+18T+30T2−702T3+8088T4+126648T5+719359T6+126648pT7+8088p2T8−702p3T9+30p4T10+18p5T11+p6T12 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12}
89 1−12T−60T2+198T3+7584T4+70800T5−1684181T6+70800pT7+7584p2T8+198p3T9−60p4T10−12p5T11+p6T12 1 - 12 T - 60 T^{2} + 198 T^{3} + 7584 T^{4} + 70800 T^{5} - 1684181 T^{6} + 70800 p T^{7} + 7584 p^{2} T^{8} + 198 p^{3} T^{9} - 60 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
97 1−3T−114T2+149T3+2421T4+11502T5+340233T6+11502pT7+2421p2T8+149p3T9−114p4T10−3p5T11+p6T12 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 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

−5.15647210108345494806655841704, −4.86380285581696211941548942749, −4.79542064840869391124347659342, −4.67000187049022545446688180544, −4.53272063276787728897164247088, −4.37733122918936761059266255226, −3.93292738038095162567008306760, −3.84639525545306640528622805239, −3.75782377640479351629457596012, −3.54060668688092052242506071789, −3.37981372863964212290991161340, −2.95406513760523355890324987096, −2.89097028496577492197810740024, −2.88553098963996835285390837704, −2.53019707865550335147930614360, −2.49642580341731286473168079976, −2.21192925779734216232361606587, −1.50691906677424877124384919067, −1.49376423775150891738678551928, −1.47004014590917633133735004636, −1.30529625134466964710113738245, −1.03744470626005648041139332905, −0.974039639536031322478803260247, −0.48897064206948034199398703299, −0.43651933964689686108467198923, 0.43651933964689686108467198923, 0.48897064206948034199398703299, 0.974039639536031322478803260247, 1.03744470626005648041139332905, 1.30529625134466964710113738245, 1.47004014590917633133735004636, 1.49376423775150891738678551928, 1.50691906677424877124384919067, 2.21192925779734216232361606587, 2.49642580341731286473168079976, 2.53019707865550335147930614360, 2.88553098963996835285390837704, 2.89097028496577492197810740024, 2.95406513760523355890324987096, 3.37981372863964212290991161340, 3.54060668688092052242506071789, 3.75782377640479351629457596012, 3.84639525545306640528622805239, 3.93292738038095162567008306760, 4.37733122918936761059266255226, 4.53272063276787728897164247088, 4.67000187049022545446688180544, 4.79542064840869391124347659342, 4.86380285581696211941548942749, 5.15647210108345494806655841704

Graph of the ZZ-function along the critical line

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