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Properties

Label 6-8512e3-1.1-c1e3-0-3
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·5-s + 3·7-s + 9-s + 7·11-s + 2·13-s + 6·15-s + 7·17-s + 3·19-s + 9·21-s − 14·23-s − 6·25-s − 7·27-s + 3·29-s + 11·31-s + 21·33-s + 6·35-s + 6·39-s − 7·41-s − 4·43-s + 2·45-s − 8·47-s + 6·49-s + 21·51-s + 53-s + 14·55-s + 9·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s + 1.13·7-s + 1/3·9-s + 2.11·11-s + 0.554·13-s + 1.54·15-s + 1.69·17-s + 0.688·19-s + 1.96·21-s − 2.91·23-s − 6/5·25-s − 1.34·27-s + 0.557·29-s + 1.97·31-s + 3.65·33-s + 1.01·35-s + 0.960·39-s − 1.09·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 6/7·49-s + 2.94·51-s + 0.137·53-s + 1.88·55-s + 1.19·57-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 18.6656015818.66560158
L(12)L(\frac12) ≈\approx 18.6656015818.66560158
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−pT+8T2−14T3+8pT4−p3T5+p3T6 1 - p T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - p^{3} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1−2T+2pT2−18T3+2p2T4−2p2T5+p3T6 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−7T+4pT2−158T3+4p2T4−7p2T5+p3T6 1 - 7 T + 4 p T^{2} - 158 T^{3} + 4 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−2T+34T2−50T3+34pT4−2p2T5+p3T6 1 - 2 T + 34 T^{2} - 50 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−7T+40T2−132T3+40pT4−7p2T5+p3T6 1 - 7 T + 40 T^{2} - 132 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+14T+122T2+700T3+122pT4+14p2T5+p3T6 1 + 14 T + 122 T^{2} + 700 T^{3} + 122 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−3T+14T2+104T3+14pT4−3p2T5+p3T6 1 - 3 T + 14 T^{2} + 104 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−11T+118T2−698T3+118pT4−11p2T5+p3T6 1 - 11 T + 118 T^{2} - 698 T^{3} + 118 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+68T2−106T3+68pT4+p3T6 1 + 68 T^{2} - 106 T^{3} + 68 p T^{4} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+7T−28T2−424T3−28pT4+7p2T5+p3T6 1 + 7 T - 28 T^{2} - 424 T^{3} - 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+4T+109T2+328T3+109pT4+4p2T5+p3T6 1 + 4 T + 109 T^{2} + 328 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+8T+112T2+768T3+112pT4+8p2T5+p3T6 1 + 8 T + 112 T^{2} + 768 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−T+128T2−104T3+128pT4−p2T5+p3T6 1 - T + 128 T^{2} - 104 T^{3} + 128 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+10T+178T2+1056T3+178pT4+10p2T5+p3T6 1 + 10 T + 178 T^{2} + 1056 T^{3} + 178 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−6T+134T2−650T3+134pT4−6p2T5+p3T6 1 - 6 T + 134 T^{2} - 650 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+3T+122T2+214T3+122pT4+3p2T5+p3T6 1 + 3 T + 122 T^{2} + 214 T^{3} + 122 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+152T2+32T3+152pT4+p3T6 1 + 152 T^{2} + 32 T^{3} + 152 p T^{4} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−T+118T2−244T3+118pT4−p2T5+p3T6 1 - T + 118 T^{2} - 244 T^{3} + 118 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−4T+193T2−664T3+193pT4−4p2T5+p3T6 1 - 4 T + 193 T^{2} - 664 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−31T+538T2−5934T3+538pT4−31p2T5+p3T6 1 - 31 T + 538 T^{2} - 5934 T^{3} + 538 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+28T+371T2+3632T3+371pT4+28p2T5+p3T6 1 + 28 T + 371 T^{2} + 3632 T^{3} + 371 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+30T+534T2+6302T3+534pT4+30p2T5+p3T6 1 + 30 T + 534 T^{2} + 6302 T^{3} + 534 p T^{4} + 30 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

−6.98641906647958439331812663765, −6.47564461825061192249005869197, −6.40408580878679048101142994072, −6.14320317469339879104189182849, −5.94484466760484790720959921545, −5.81191300851102184630777805351, −5.70275178498974691265615399924, −5.07878989057910058875472338922, −5.00528129767450521625682881655, −4.82218904940168473079462875510, −4.39176348779579402765392290694, −4.13092749297459335052364339505, −3.92716281520920824549684092588, −3.57225496050796564723710436144, −3.45596550103930812981658628031, −3.42389240255062165013585662218, −2.82885643283701257055206870020, −2.60571594550086436799369276668, −2.52028471541153330313171010601, −1.77652584675795969726433162758, −1.76490806364060081069254543422, −1.72078824840405871527472892168, −1.39395629877196249280075932145, −0.77583049646881503805110798565, −0.52730190583695110422846800207, 0.52730190583695110422846800207, 0.77583049646881503805110798565, 1.39395629877196249280075932145, 1.72078824840405871527472892168, 1.76490806364060081069254543422, 1.77652584675795969726433162758, 2.52028471541153330313171010601, 2.60571594550086436799369276668, 2.82885643283701257055206870020, 3.42389240255062165013585662218, 3.45596550103930812981658628031, 3.57225496050796564723710436144, 3.92716281520920824549684092588, 4.13092749297459335052364339505, 4.39176348779579402765392290694, 4.82218904940168473079462875510, 5.00528129767450521625682881655, 5.07878989057910058875472338922, 5.70275178498974691265615399924, 5.81191300851102184630777805351, 5.94484466760484790720959921545, 6.14320317469339879104189182849, 6.40408580878679048101142994072, 6.47564461825061192249005869197, 6.98641906647958439331812663765

Graph of the ZZ-function along the critical line