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Properties

Label 6-8512e3-1.1-c1e3-0-1
Degree 66
Conductor 616729673728616729673728
Sign 11
Analytic cond. 313997.313997.
Root an. cond. 8.244318.24431
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  + 3-s − 2·5-s − 3·7-s − 9-s + 3·11-s − 8·13-s − 2·15-s + 9·17-s + 3·19-s − 3·21-s + 4·23-s − 2·25-s − 5·27-s − 11·29-s − 5·31-s + 3·33-s + 6·35-s − 4·37-s − 8·39-s + 11·41-s + 4·43-s + 2·45-s − 2·47-s + 6·49-s + 9·51-s − 9·53-s − 6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.13·7-s − 1/3·9-s + 0.904·11-s − 2.21·13-s − 0.516·15-s + 2.18·17-s + 0.688·19-s − 0.654·21-s + 0.834·23-s − 2/5·25-s − 0.962·27-s − 2.04·29-s − 0.898·31-s + 0.522·33-s + 1.01·35-s − 0.657·37-s − 1.28·39-s + 1.71·41-s + 0.609·43-s + 0.298·45-s − 0.291·47-s + 6/7·49-s + 1.26·51-s − 1.23·53-s − 0.809·55-s + ⋯

Functional equation

Λ(s)=((218⋅73⋅193)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅73⋅193)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{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: 218⋅73⋅1932^{18} \cdot 7^{3} \cdot 19^{3}
Sign: 11
Analytic conductor: 313997.313997.
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 218⋅73⋅193, ( :1/2,1/2,1/2), 1)(6,\ 2^{18} \cdot 7^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.3406539381.340653938
L(12)L(\frac12) ≈\approx 1.3406539381.340653938
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
7C1C_1 (1+T)3 ( 1 + T )^{3}
19C1C_1 (1−T)3 ( 1 - T )^{3}
good3S4×C2S_4\times C_2 1−T+2T2+2T3+2pT4−p2T5+p3T6 1 - T + 2 T^{2} + 2 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1+2T+6T2+6T3+6pT4+2p2T5+p3T6 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−3T+26T2−46T3+26pT4−3p2T5+p3T6 1 - 3 T + 26 T^{2} - 46 T^{3} + 26 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+8T+50T2+192T3+50pT4+8p2T5+p3T6 1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−9T+4pT2−292T3+4p2T4−9p2T5+p3T6 1 - 9 T + 4 p T^{2} - 292 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−4T+64T2−180T3+64pT4−4p2T5+p3T6 1 - 4 T + 64 T^{2} - 180 T^{3} + 64 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+11T+102T2+636T3+102pT4+11p2T5+p3T6 1 + 11 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1+5T+76T2+314T3+76pT4+5p2T5+p3T6 1 + 5 T + 76 T^{2} + 314 T^{3} + 76 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+4T+92T2+246T3+92pT4+4p2T5+p3T6 1 + 4 T + 92 T^{2} + 246 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−11T+110T2−622T3+110pT4−11p2T5+p3T6 1 - 11 T + 110 T^{2} - 622 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−4T+37T2−376T3+37pT4−4p2T5+p3T6 1 - 4 T + 37 T^{2} - 376 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+2T+66T2+372T3+66pT4+2p2T5+p3T6 1 + 2 T + 66 T^{2} + 372 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+9T+120T2+632T3+120pT4+9p2T5+p3T6 1 + 9 T + 120 T^{2} + 632 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−12T+132T2−1308T3+132pT4−12p2T5+p3T6 1 - 12 T + 132 T^{2} - 1308 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−2T+26T2−42T3+26pT4−2p2T5+p3T6 1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−9T+218T2−1192T3+218pT4−9p2T5+p3T6 1 - 9 T + 218 T^{2} - 1192 T^{3} + 218 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+158T2−58T3+158pT4+p3T6 1 + 158 T^{2} - 58 T^{3} + 158 p T^{4} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+21T+302T2+2764T3+302pT4+21p2T5+p3T6 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−6T+89T2+68T3+89pT4−6p2T5+p3T6 1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−7T+204T2−1062T3+204pT4−7p2T5+p3T6 1 - 7 T + 204 T^{2} - 1062 T^{3} + 204 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+18T+335T2+3268T3+335pT4+18p2T5+p3T6 1 + 18 T + 335 T^{2} + 3268 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−28T+330T2−2896T3+330pT4−28p2T5+p3T6 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 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

−7.19181693558475457057465620795, −6.61487667750844684262870826751, −6.46697120505320068380294690365, −6.21304288510182901975075719130, −5.83736391754173069462216679333, −5.66612198719148753506840354935, −5.60194634240205233720156755742, −5.17926863514407035360079750163, −5.07091069185580242922509838651, −4.85349783164951712195728069608, −4.35055237876135894059879165030, −4.20890690505100790948186631897, −3.72319369770100674091808217261, −3.71894719284193403835209244211, −3.62009514222440069080929156965, −3.24900645801641252767418413092, −2.95417738949078819218298199656, −2.71557733646255377679495739498, −2.54311832246423664398305855522, −1.97293460850486025101056358784, −1.96536593976643843346974860924, −1.34979862011024683108000362082, −1.10395185667699594294475763082, −0.52458235259035336005619217026, −0.24985435976138412993749454907, 0.24985435976138412993749454907, 0.52458235259035336005619217026, 1.10395185667699594294475763082, 1.34979862011024683108000362082, 1.96536593976643843346974860924, 1.97293460850486025101056358784, 2.54311832246423664398305855522, 2.71557733646255377679495739498, 2.95417738949078819218298199656, 3.24900645801641252767418413092, 3.62009514222440069080929156965, 3.71894719284193403835209244211, 3.72319369770100674091808217261, 4.20890690505100790948186631897, 4.35055237876135894059879165030, 4.85349783164951712195728069608, 5.07091069185580242922509838651, 5.17926863514407035360079750163, 5.60194634240205233720156755742, 5.66612198719148753506840354935, 5.83736391754173069462216679333, 6.21304288510182901975075719130, 6.46697120505320068380294690365, 6.61487667750844684262870826751, 7.19181693558475457057465620795

Graph of the ZZ-function along the critical line