↑

Properties

Label 6-8112e3-1.1-c1e3-0-16
Degree 66
Conductor 533806460928533806460928
Sign −1-1
Analytic cond. 271778.271778.
Root an. cond. 8.048268.04826
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·5-s + 10·7-s + 6·9-s − 11-s + 12·15-s − 7·17-s + 11·19-s − 30·21-s − 2·23-s − 2·25-s − 10·27-s − 8·29-s + 8·31-s + 3·33-s − 40·35-s − 14·37-s − 41-s + 3·43-s − 24·45-s − 9·47-s + 48·49-s + 21·51-s − 13·53-s + 4·55-s − 33·57-s − 14·59-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s + 3.77·7-s + 2·9-s − 0.301·11-s + 3.09·15-s − 1.69·17-s + 2.52·19-s − 6.54·21-s − 0.417·23-s − 2/5·25-s − 1.92·27-s − 1.48·29-s + 1.43·31-s + 0.522·33-s − 6.76·35-s − 2.30·37-s − 0.156·41-s + 0.457·43-s − 3.57·45-s − 1.31·47-s + 48/7·49-s + 2.94·51-s − 1.78·53-s + 0.539·55-s − 4.37·57-s − 1.82·59-s + ⋯

Functional equation

Λ(s)=((212⋅33⋅136)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅33⋅136)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 212⋅33⋅1362^{12} \cdot 3^{3} \cdot 13^{6}
Sign: −1-1
Analytic conductor: 271778.271778.
Root analytic conductor: 8.048268.04826
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 212⋅33⋅136, ( :1/2,1/2,1/2), −1)(6,\ 2^{12} \cdot 3^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1+T)3 ( 1 + T )^{3}
13 1 1
good5A4×C2A_4\times C_2 1+4T+18T2+39T3+18pT4+4p2T5+p3T6 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
7A4×C2A_4\times C_2 1−10T+52T2−169T3+52pT4−10p2T5+p3T6 1 - 10 T + 52 T^{2} - 169 T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
11A4×C2A_4\times C_2 1+T+3T2−21T3+3pT4+p2T5+p3T6 1 + T + 3 T^{2} - 21 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
17A4×C2A_4\times C_2 1+7T+65T2+245T3+65pT4+7p2T5+p3T6 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
19A4×C2A_4\times C_2 1−11T+67T2−305T3+67pT4−11p2T5+p3T6 1 - 11 T + 67 T^{2} - 305 T^{3} + 67 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}
23A4×C2A_4\times C_2 1+2T+26T2+175T3+26pT4+2p2T5+p3T6 1 + 2 T + 26 T^{2} + 175 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
29A4×C2A_4\times C_2 1+8T+92T2+421T3+92pT4+8p2T5+p3T6 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
31A4×C2A_4\times C_2 1−8T+70T2−299T3+70pT4−8p2T5+p3T6 1 - 8 T + 70 T^{2} - 299 T^{3} + 70 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
37A4×C2A_4\times C_2 1+14T+174T2+1127T3+174pT4+14p2T5+p3T6 1 + 14 T + 174 T^{2} + 1127 T^{3} + 174 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}
41A4×C2A_4\times C_2 1+T+121T2+81T3+121pT4+p2T5+p3T6 1 + T + 121 T^{2} + 81 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
43A4×C2A_4\times C_2 1−3T+104T2−287T3+104pT4−3p2T5+p3T6 1 - 3 T + 104 T^{2} - 287 T^{3} + 104 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
47A4×C2A_4\times C_2 1+9T+21T2−65T3+21pT4+9p2T5+p3T6 1 + 9 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
53A4×C2A_4\times C_2 1+13T+199T2+1407T3+199pT4+13p2T5+p3T6 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
59A4×C2A_4\times C_2 1+14T+3pT2+1596T3+3p2T4+14p2T5+p3T6 1 + 14 T + 3 p T^{2} + 1596 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}
61A4×C2A_4\times C_2 1+13T+195T2+1363T3+195pT4+13p2T5+p3T6 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
67A4×C2A_4\times C_2 1−5T+179T2−573T3+179pT4−5p2T5+p3T6 1 - 5 T + 179 T^{2} - 573 T^{3} + 179 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
71A4×C2A_4\times C_2 1+6T+134T2+391T3+134pT4+6p2T5+p3T6 1 + 6 T + 134 T^{2} + 391 T^{3} + 134 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
73A4×C2A_4\times C_2 1+18T+320T2+2795T3+320pT4+18p2T5+p3T6 1 + 18 T + 320 T^{2} + 2795 T^{3} + 320 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
79A4×C2A_4\times C_2 1−9T+215T2−1253T3+215pT4−9p2T5+p3T6 1 - 9 T + 215 T^{2} - 1253 T^{3} + 215 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
83A4×C2A_4\times C_2 1+16T+304T2+2613T3+304pT4+16p2T5+p3T6 1 + 16 T + 304 T^{2} + 2613 T^{3} + 304 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}
89A4×C2A_4\times C_2 1−5T+259T2−891T3+259pT4−5p2T5+p3T6 1 - 5 T + 259 T^{2} - 891 T^{3} + 259 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
97A4×C2A_4\times C_2 1+5T+122T2−219T3+122pT4+5p2T5+p3T6 1 + 5 T + 122 T^{2} - 219 T^{3} + 122 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
show more
show less
   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.46836246232969281859259778709, −7.15499357234526488646541123102, −6.88366889187667061264006806164, −6.63807028833746263630585631779, −6.14100759199415691748954335292, −6.14029808428758175069274343325, −5.73760790204501815927437412603, −5.48603547886983487259632495454, −5.39453012248102383284089808500, −4.96711698134447443500017922298, −4.83218458492577701461055795075, −4.83159231670333882456599014597, −4.55014710503690894056304793394, −4.16655739806712117005005788203, −4.09647229806733100431844738850, −4.02310379444992843536757895038, −3.23471640339378557341973275944, −3.18866982013216305274387279240, −3.13246078384785048990267992256, −2.14690901412227149522107149218, −2.08301381156695593002848589701, −1.85340977498473678510209724924, −1.43487749347466136276338454014, −1.16178119709563314028245008067, −1.15972287209929871956668103191, 0, 0, 0, 1.15972287209929871956668103191, 1.16178119709563314028245008067, 1.43487749347466136276338454014, 1.85340977498473678510209724924, 2.08301381156695593002848589701, 2.14690901412227149522107149218, 3.13246078384785048990267992256, 3.18866982013216305274387279240, 3.23471640339378557341973275944, 4.02310379444992843536757895038, 4.09647229806733100431844738850, 4.16655739806712117005005788203, 4.55014710503690894056304793394, 4.83159231670333882456599014597, 4.83218458492577701461055795075, 4.96711698134447443500017922298, 5.39453012248102383284089808500, 5.48603547886983487259632495454, 5.73760790204501815927437412603, 6.14029808428758175069274343325, 6.14100759199415691748954335292, 6.63807028833746263630585631779, 6.88366889187667061264006806164, 7.15499357234526488646541123102, 7.46836246232969281859259778709

Graph of the ZZ-function along the critical line