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Properties

Label 6-760e3-1.1-c1e3-0-2
Degree 66
Conductor 438976000438976000
Sign −1-1
Analytic cond. 223.497223.497
Root an. cond. 2.463452.46345
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s − 4·9-s − 11·13-s + 3·15-s − 3·17-s + 3·19-s + 21-s − 9·23-s + 6·25-s + 5·27-s − 7·29-s + 6·31-s + 3·35-s − 20·37-s + 11·39-s − 22·41-s − 10·43-s + 12·45-s − 4·49-s + 3·51-s − 7·53-s − 3·57-s + 11·59-s − 16·61-s + 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s − 4/3·9-s − 3.05·13-s + 0.774·15-s − 0.727·17-s + 0.688·19-s + 0.218·21-s − 1.87·23-s + 6/5·25-s + 0.962·27-s − 1.29·29-s + 1.07·31-s + 0.507·35-s − 3.28·37-s + 1.76·39-s − 3.43·41-s − 1.52·43-s + 1.78·45-s − 4/7·49-s + 0.420·51-s − 0.961·53-s − 0.397·57-s + 1.43·59-s − 2.04·61-s + 0.503·63-s + ⋯

Functional equation

Λ(s)=((29⋅53⋅193)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((29⋅53⋅193)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 29⋅53⋅1932^{9} \cdot 5^{3} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 223.497223.497
Root analytic conductor: 2.463452.46345
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 29⋅53⋅193, ( :1/2,1/2,1/2), −1)(6,\ 2^{9} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 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
5C1C_1 (1+T)3 ( 1 + T )^{3}
19C1C_1 (1−T)3 ( 1 - T )^{3}
good3S4×C2S_4\times C_2 1+T+5T2+4T3+5pT4+p2T5+p3T6 1 + T + 5 T^{2} + 4 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+T+5T2+30T3+5pT4+p2T5+p3T6 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+5T2+16T3+5pT4+p3T6 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+11T+55T2+200T3+55pT4+11p2T5+p3T6 1 + 11 T + 55 T^{2} + 200 T^{3} + 55 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+3T+47T2+98T3+47pT4+3p2T5+p3T6 1 + 3 T + 47 T^{2} + 98 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+9T+89T2+422T3+89pT4+9p2T5+p3T6 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+7T+43T2+114T3+43pT4+7p2T5+p3T6 1 + 7 T + 43 T^{2} + 114 T^{3} + 43 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−6T+77T2−340T3+77pT4−6p2T5+p3T6 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+20T+237T2+1724T3+237pT4+20p2T5+p3T6 1 + 20 T + 237 T^{2} + 1724 T^{3} + 237 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}
41D6D_{6} 1+22T+223T2+1572T3+223pT4+22p2T5+p3T6 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+10T+97T2+508T3+97pT4+10p2T5+p3T6 1 + 10 T + 97 T^{2} + 508 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+29T2−128T3+29pT4+p3T6 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+7T−9T2−600T3−9pT4+7p2T5+p3T6 1 + 7 T - 9 T^{2} - 600 T^{3} - 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−11T+37T2+246T3+37pT4−11p2T5+p3T6 1 - 11 T + 37 T^{2} + 246 T^{3} + 37 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+16T+207T2+1600T3+207pT4+16p2T5+p3T6 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+T+101T2+396T3+101pT4+p2T5+p3T6 1 + T + 101 T^{2} + 396 T^{3} + 101 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
71C2C_2 (1+pT2)3 ( 1 + p T^{2} )^{3}
73S4×C2S_4\times C_2 1+5T+47T2−498T3+47pT4+5p2T5+p3T6 1 + 5 T + 47 T^{2} - 498 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−26T+445T2−4604T3+445pT4−26p2T5+p3T6 1 - 26 T + 445 T^{2} - 4604 T^{3} + 445 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+14T+297T2+2340T3+297pT4+14p2T5+p3T6 1 + 14 T + 297 T^{2} + 2340 T^{3} + 297 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+6T+15T2−188T3+15pT4+6p2T5+p3T6 1 + 6 T + 15 T^{2} - 188 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−8T+233T2−1260T3+233pT4−8p2T5+p3T6 1 - 8 T + 233 T^{2} - 1260 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
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   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

−9.851047708190152447872137914912, −9.300111935762805548800554094466, −9.082066331624621691716459984824, −8.737575313337994366301470568337, −8.320629452269819123312144686836, −8.228861095782982207773762068028, −8.012238407729892217142472895756, −7.53096182092576360959363346625, −7.24289138361354677131164445580, −7.11895386741679432208888755894, −6.69824582957521889411372400506, −6.42565122341162467266792565573, −6.26440952286459499269251298369, −5.42176104799751120212455301215, −5.31852128526798038013231331601, −5.27819174808657037261681585591, −4.71825770333991849318671689267, −4.62080971355182172365566202781, −4.07072846440327463695774985273, −3.51360915669260130052113253599, −3.25415039825249590894571527107, −3.15140318969988800211125564551, −2.46246116481331798869274685032, −2.02150641146546994728047556882, −1.66190957001146624107476081496, 0, 0, 0, 1.66190957001146624107476081496, 2.02150641146546994728047556882, 2.46246116481331798869274685032, 3.15140318969988800211125564551, 3.25415039825249590894571527107, 3.51360915669260130052113253599, 4.07072846440327463695774985273, 4.62080971355182172365566202781, 4.71825770333991849318671689267, 5.27819174808657037261681585591, 5.31852128526798038013231331601, 5.42176104799751120212455301215, 6.26440952286459499269251298369, 6.42565122341162467266792565573, 6.69824582957521889411372400506, 7.11895386741679432208888755894, 7.24289138361354677131164445580, 7.53096182092576360959363346625, 8.012238407729892217142472895756, 8.228861095782982207773762068028, 8.320629452269819123312144686836, 8.737575313337994366301470568337, 9.082066331624621691716459984824, 9.300111935762805548800554094466, 9.851047708190152447872137914912

Graph of the ZZ-function along the critical line