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Properties

Label 6-4788e3-1.1-c1e3-0-1
Degree 66
Conductor 109764631872109764631872
Sign −1-1
Analytic cond. 55884.855884.8
Root an. cond. 6.183236.18323
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  − 2·5-s − 3·7-s + 3·11-s + 8·13-s − 9·17-s − 3·19-s − 4·23-s − 2·25-s − 11·29-s − 5·31-s + 6·35-s + 4·37-s − 11·41-s − 4·43-s + 2·47-s + 6·49-s − 9·53-s − 6·55-s + 12·59-s − 2·61-s − 16·65-s − 9·67-s − 21·73-s − 9·77-s + 6·79-s + 7·83-s + 18·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.13·7-s + 0.904·11-s + 2.21·13-s − 2.18·17-s − 0.688·19-s − 0.834·23-s − 2/5·25-s − 2.04·29-s − 0.898·31-s + 1.01·35-s + 0.657·37-s − 1.71·41-s − 0.609·43-s + 0.291·47-s + 6/7·49-s − 1.23·53-s − 0.809·55-s + 1.56·59-s − 0.256·61-s − 1.98·65-s − 1.09·67-s − 2.45·73-s − 1.02·77-s + 0.675·79-s + 0.768·83-s + 1.95·85-s + ⋯

Functional equation

Λ(s)=((26⋅36⋅73⋅193)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅36⋅73⋅193)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \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: 26⋅36⋅73⋅1932^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 55884.855884.8
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 26⋅36⋅73⋅193, ( :1/2,1/2,1/2), −1)(6,\ 2^{6} \cdot 3^{6} \cdot 7^{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
3 1 1
7C1C_1 (1+T)3 ( 1 + T )^{3}
19C1C_1 (1+T)3 ( 1 + T )^{3}
good5S4×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

−7.80233357009246873797216254831, −7.45709637976400741588544457565, −7.05525600287532077528601368695, −7.01460571540713490223087234466, −6.58981154981396501174413316534, −6.48949630681350100625244847123, −6.35892855877094674435060863559, −5.89121501983364901012169717262, −5.87936516085587305840767800565, −5.81180814346290804813736336848, −5.08894741655729006122253346542, −4.84329458615620407145141925152, −4.83133918485081641807896272980, −4.14156748431979660425516907669, −4.01055706408249825884400235926, −3.95715600855877940973954997520, −3.61417168403294593911068923119, −3.48509581821501164855341846140, −3.33193055776557401191519542690, −2.63091160861477335470633531621, −2.40920811542236548713162993931, −2.12635607869483977801507754761, −1.71142352880627929744876132350, −1.32804388299010437186613276852, −1.14207886106608566229236838372, 0, 0, 0, 1.14207886106608566229236838372, 1.32804388299010437186613276852, 1.71142352880627929744876132350, 2.12635607869483977801507754761, 2.40920811542236548713162993931, 2.63091160861477335470633531621, 3.33193055776557401191519542690, 3.48509581821501164855341846140, 3.61417168403294593911068923119, 3.95715600855877940973954997520, 4.01055706408249825884400235926, 4.14156748431979660425516907669, 4.83133918485081641807896272980, 4.84329458615620407145141925152, 5.08894741655729006122253346542, 5.81180814346290804813736336848, 5.87936516085587305840767800565, 5.89121501983364901012169717262, 6.35892855877094674435060863559, 6.48949630681350100625244847123, 6.58981154981396501174413316534, 7.01460571540713490223087234466, 7.05525600287532077528601368695, 7.45709637976400741588544457565, 7.80233357009246873797216254831

Graph of the ZZ-function along the critical line