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Properties

Label 6-3888e3-1.1-c1e3-0-5
Degree 66
Conductor 5877312307258773123072
Sign −1-1
Analytic cond. 29923.329923.3
Root an. cond. 5.571875.57187
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  − 6·5-s + 3·7-s + 3·11-s − 3·13-s − 9·17-s + 3·19-s + 6·23-s + 12·25-s − 12·29-s + 12·31-s − 18·35-s − 3·37-s + 3·41-s + 12·43-s − 6·47-s − 6·49-s − 18·53-s − 18·55-s − 21·59-s + 6·61-s + 18·65-s − 6·67-s − 9·71-s + 6·73-s + 9·77-s − 6·79-s − 6·83-s + ⋯
L(s)  = 1  − 2.68·5-s + 1.13·7-s + 0.904·11-s − 0.832·13-s − 2.18·17-s + 0.688·19-s + 1.25·23-s + 12/5·25-s − 2.22·29-s + 2.15·31-s − 3.04·35-s − 0.493·37-s + 0.468·41-s + 1.82·43-s − 0.875·47-s − 6/7·49-s − 2.47·53-s − 2.42·55-s − 2.73·59-s + 0.768·61-s + 2.23·65-s − 0.733·67-s − 1.06·71-s + 0.702·73-s + 1.02·77-s − 0.675·79-s − 0.658·83-s + ⋯

Functional equation

Λ(s)=((212⋅315)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅315)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 212⋅3152^{12} \cdot 3^{15}
Sign: −1-1
Analytic conductor: 29923.329923.3
Root analytic conductor: 5.571875.57187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 212⋅315, ( :1/2,1/2,1/2), −1)(6,\ 2^{12} \cdot 3^{15} ,\ ( \ : 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
3 1 1
good5A4×C2A_4\times C_2 1+6T+24T2+63T3+24pT4+6p2T5+p3T6 1 + 6 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
7A4×C2A_4\times C_2 1−3T+15T2−25T3+15pT4−3p2T5+p3T6 1 - 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
11A4×C2A_4\times C_2 1−3T+15T2−63T3+15pT4−3p2T5+p3T6 1 - 3 T + 15 T^{2} - 63 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
13A4×C2A_4\times C_2 1+3T+33T2+61T3+33pT4+3p2T5+p3T6 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
17C2C_2 (1+3T+pT2)3 ( 1 + 3 T + p T^{2} )^{3}
19A4×C2A_4\times C_2 1−3T+33T2−115T3+33pT4−3p2T5+p3T6 1 - 3 T + 33 T^{2} - 115 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
23A4×C2A_4\times C_2 1−6T+60T2−225T3+60pT4−6p2T5+p3T6 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
29A4×C2A_4\times C_2 1+12T+114T2+639T3+114pT4+12p2T5+p3T6 1 + 12 T + 114 T^{2} + 639 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
31A4×C2A_4\times C_2 1−12T+132T2−763T3+132pT4−12p2T5+p3T6 1 - 12 T + 132 T^{2} - 763 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
37A4×C2A_4\times C_2 1+3T+87T2+223T3+87pT4+3p2T5+p3T6 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
41A4×C2A_4\times C_2 1−3T+69T2−27T3+69pT4−3p2T5+p3T6 1 - 3 T + 69 T^{2} - 27 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
43A4×C2A_4\times C_2 1−12T+168T2−1051T3+168pT4−12p2T5+p3T6 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
47A4×C2A_4\times C_2 1+6T+78T2+297T3+78pT4+6p2T5+p3T6 1 + 6 T + 78 T^{2} + 297 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
53A4×C2A_4\times C_2 1+18T+240T2+1989T3+240pT4+18p2T5+p3T6 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
59A4×C2A_4\times C_2 1+21T+321T2+2799T3+321pT4+21p2T5+p3T6 1 + 21 T + 321 T^{2} + 2799 T^{3} + 321 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}
61A4×C2A_4\times C_2 1−6T+132T2−785T3+132pT4−6p2T5+p3T6 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
67A4×C2A_4\times C_2 1+6T+150T2+695T3+150pT4+6p2T5+p3T6 1 + 6 T + 150 T^{2} + 695 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
71A4×C2A_4\times C_2 1+9T+51T2+279T3+51pT4+9p2T5+p3T6 1 + 9 T + 51 T^{2} + 279 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
73A4×C2A_4\times C_2 1−6T+150T2−479T3+150pT4−6p2T5+p3T6 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
79A4×C2A_4\times C_2 1+6T+186T2+1001T3+186pT4+6p2T5+p3T6 1 + 6 T + 186 T^{2} + 1001 T^{3} + 186 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
83A4×C2A_4\times C_2 1+6T+222T2+945T3+222pT4+6p2T5+p3T6 1 + 6 T + 222 T^{2} + 945 T^{3} + 222 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
89A4×C2A_4\times C_2 1+78T2−999T3+78pT4+p3T6 1 + 78 T^{2} - 999 T^{3} + 78 p T^{4} + p^{3} T^{6}
97A4×C2A_4\times C_2 1−15T+222T2−2891T3+222pT4−15p2T5+p3T6 1 - 15 T + 222 T^{2} - 2891 T^{3} + 222 p T^{4} - 15 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

−7.949006234235659012259460467551, −7.61900457199935498004591451961, −7.47121996593214277545433940460, −7.36901086495246683706311938885, −6.85504095659697584647232344179, −6.73255519659357397100650458680, −6.62881203407608945484571557486, −6.10289652796523199982805761756, −5.89022826964615820466444961335, −5.77314750969281902168893971711, −4.99473379624509408119723929810, −4.85510581554273203279306503146, −4.83341215912606419559855530729, −4.62926178415234484386524268833, −4.23953675827492579621695492413, −4.10319146589635572577495966231, −3.65736091024836392054639909659, −3.57230502601571549533672351001, −3.37191862324751162921187324734, −2.62758414343559060131625559676, −2.57177888177779657641788038413, −2.41520092911475986425798225864, −1.53061570986950636523135884593, −1.30793952740054271621150881334, −1.28298563183084351914027469446, 0, 0, 0, 1.28298563183084351914027469446, 1.30793952740054271621150881334, 1.53061570986950636523135884593, 2.41520092911475986425798225864, 2.57177888177779657641788038413, 2.62758414343559060131625559676, 3.37191862324751162921187324734, 3.57230502601571549533672351001, 3.65736091024836392054639909659, 4.10319146589635572577495966231, 4.23953675827492579621695492413, 4.62926178415234484386524268833, 4.83341215912606419559855530729, 4.85510581554273203279306503146, 4.99473379624509408119723929810, 5.77314750969281902168893971711, 5.89022826964615820466444961335, 6.10289652796523199982805761756, 6.62881203407608945484571557486, 6.73255519659357397100650458680, 6.85504095659697584647232344179, 7.36901086495246683706311938885, 7.47121996593214277545433940460, 7.61900457199935498004591451961, 7.949006234235659012259460467551

Graph of the ZZ-function along the critical line