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Properties

Label 6-165e3-1.1-c3e3-0-2
Degree 66
Conductor 44921254492125
Sign −1-1
Analytic cond. 922.677922.677
Root an. cond. 3.120143.12014
Motivic weight 33
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  − 4·2-s − 9·3-s + 7·4-s − 15·5-s + 36·6-s − 4·7-s − 12·8-s + 54·9-s + 60·10-s + 33·11-s − 63·12-s + 16·14-s + 135·15-s − 7·16-s − 218·17-s − 216·18-s + 146·19-s − 105·20-s + 36·21-s − 132·22-s − 200·23-s + 108·24-s + 150·25-s − 270·27-s − 28·28-s + 68·29-s − 540·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 7/8·4-s − 1.34·5-s + 2.44·6-s − 0.215·7-s − 0.530·8-s + 2·9-s + 1.89·10-s + 0.904·11-s − 1.51·12-s + 0.305·14-s + 2.32·15-s − 0.109·16-s − 3.11·17-s − 2.82·18-s + 1.76·19-s − 1.17·20-s + 0.374·21-s − 1.27·22-s − 1.81·23-s + 0.918·24-s + 6/5·25-s − 1.92·27-s − 0.188·28-s + 0.435·29-s − 3.28·30-s + ⋯

Functional equation

Λ(s)=(4492125s/2 ΓC(s)3 L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(4492125s/2 ΓC(s+3/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 66
Conductor: 44921254492125    =    33⋅53⋅1133^{3} \cdot 5^{3} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 922.677922.677
Root analytic conductor: 3.120143.12014
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 4492125, ( :3/2,3/2,3/2), −1)(6,\ 4492125,\ (\ :3/2, 3/2, 3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3C1C_1 (1+pT)3 ( 1 + p T )^{3}
5C1C_1 (1+pT)3 ( 1 + p T )^{3}
11C1C_1 (1−pT)3 ( 1 - p T )^{3}
good2S4×C2S_4\times C_2 1+p2T+9T2+5p2T3+9p3T4+p8T5+p9T6 1 + p^{2} T + 9 T^{2} + 5 p^{2} T^{3} + 9 p^{3} T^{4} + p^{8} T^{5} + p^{9} T^{6}
7S4×C2S_4\times C_2 1+4T+601T2−1000T3+601p3T4+4p6T5+p9T6 1 + 4 T + 601 T^{2} - 1000 T^{3} + 601 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6}
13S4×C2S_4\times C_2 1+3031T2−34144T3+3031p3T4+p9T6 1 + 3031 T^{2} - 34144 T^{3} + 3031 p^{3} T^{4} + p^{9} T^{6}
17S4×C2S_4\times C_2 1+218T+24419T2+1906964T3+24419p3T4+218p6T5+p9T6 1 + 218 T + 24419 T^{2} + 1906964 T^{3} + 24419 p^{3} T^{4} + 218 p^{6} T^{5} + p^{9} T^{6}
19S4×C2S_4\times C_2 1−146T+25953T2−2033788T3+25953p3T4−146p6T5+p9T6 1 - 146 T + 25953 T^{2} - 2033788 T^{3} + 25953 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6}
23S4×C2S_4\times C_2 1+200T+33829T2+4868464T3+33829p3T4+200p6T5+p9T6 1 + 200 T + 33829 T^{2} + 4868464 T^{3} + 33829 p^{3} T^{4} + 200 p^{6} T^{5} + p^{9} T^{6}
29S4×C2S_4\times C_2 1−68T+18803T2−153848T3+18803p3T4−68p6T5+p9T6 1 - 68 T + 18803 T^{2} - 153848 T^{3} + 18803 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6}
31S4×C2S_4\times C_2 1+68T+66141T2+2239480T3+66141p3T4+68p6T5+p9T6 1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6}
37S4×C2S_4\times C_2 1+390T+188419T2+40128292T3+188419p3T4+390p6T5+p9T6 1 + 390 T + 188419 T^{2} + 40128292 T^{3} + 188419 p^{3} T^{4} + 390 p^{6} T^{5} + p^{9} T^{6}
41S4×C2S_4\times C_2 1+196T+4407pT2+22652824T3+4407p4T4+196p6T5+p9T6 1 + 196 T + 4407 p T^{2} + 22652824 T^{3} + 4407 p^{4} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6}
43S4×C2S_4\times C_2 1+524T+209853T2+52049416T3+209853p3T4+524p6T5+p9T6 1 + 524 T + 209853 T^{2} + 52049416 T^{3} + 209853 p^{3} T^{4} + 524 p^{6} T^{5} + p^{9} T^{6}
47S4×C2S_4\times C_2 1+60T+175549T2−8508216T3+175549p3T4+60p6T5+p9T6 1 + 60 T + 175549 T^{2} - 8508216 T^{3} + 175549 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6}
53S4×C2S_4\times C_2 1+158T+185779T2+86620084T3+185779p3T4+158p6T5+p9T6 1 + 158 T + 185779 T^{2} + 86620084 T^{3} + 185779 p^{3} T^{4} + 158 p^{6} T^{5} + p^{9} T^{6}
59S4×C2S_4\times C_2 1+1044T+680281T2+344604088T3+680281p3T4+1044p6T5+p9T6 1 + 1044 T + 680281 T^{2} + 344604088 T^{3} + 680281 p^{3} T^{4} + 1044 p^{6} T^{5} + p^{9} T^{6}
61S4×C2S_4\times C_2 1−642T+620395T2−268686220T3+620395p3T4−642p6T5+p9T6 1 - 642 T + 620395 T^{2} - 268686220 T^{3} + 620395 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6}
67S4×C2S_4\times C_2 1+236T+440081T2+229497800T3+440081p3T4+236p6T5+p9T6 1 + 236 T + 440081 T^{2} + 229497800 T^{3} + 440081 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6}
71S4×C2S_4\times C_2 1+544T+944005T2+395960768T3+944005p3T4+544p6T5+p9T6 1 + 544 T + 944005 T^{2} + 395960768 T^{3} + 944005 p^{3} T^{4} + 544 p^{6} T^{5} + p^{9} T^{6}
73S4×C2S_4\times C_2 1−900T+867475T2−705839944T3+867475p3T4−900p6T5+p9T6 1 - 900 T + 867475 T^{2} - 705839944 T^{3} + 867475 p^{3} T^{4} - 900 p^{6} T^{5} + p^{9} T^{6}
79S4×C2S_4\times C_2 1+1586T+2041325T2+1549224716T3+2041325p3T4+1586p6T5+p9T6 1 + 1586 T + 2041325 T^{2} + 1549224716 T^{3} + 2041325 p^{3} T^{4} + 1586 p^{6} T^{5} + p^{9} T^{6}
83S4×C2S_4\times C_2 1+1582T+1407717T2+884488684T3+1407717p3T4+1582p6T5+p9T6 1 + 1582 T + 1407717 T^{2} + 884488684 T^{3} + 1407717 p^{3} T^{4} + 1582 p^{6} T^{5} + p^{9} T^{6}
89S4×C2S_4\times C_2 1+2122T+3521847T2+3285333068T3+3521847p3T4+2122p6T5+p9T6 1 + 2122 T + 3521847 T^{2} + 3285333068 T^{3} + 3521847 p^{3} T^{4} + 2122 p^{6} T^{5} + p^{9} T^{6}
97S4×C2S_4\times C_2 1−618T+1446319T2−1351607564T3+1446319p3T4−618p6T5+p9T6 1 - 618 T + 1446319 T^{2} - 1351607564 T^{3} + 1446319 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} 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

−11.38396107918053199134896155671, −11.31503634546292837979045754117, −11.06426070673374402137520530015, −10.62572089363730959095947307234, −9.987695812572941867857313303660, −9.854746844123005947391193913803, −9.784012941071971075327506411725, −8.883988044396467932812105532638, −8.822250236537784233102160049972, −8.712145943629755070877300170293, −7.901200733731107632500869835234, −7.87615370331142558249828812588, −7.11217076462862682062123649881, −6.82088248415705226338381828771, −6.78837948449091974826726938817, −6.33632740601128452448267112696, −5.88239134962647394570004081477, −5.06730892068729817480269668819, −5.05832509076758630826783281233, −4.23912785895003769931341466511, −4.15210514423586779419605478593, −3.50437855314430610577113437027, −2.79927966384044821933287276777, −1.66260061704225131346789765018, −1.46313092156811801131018048418, 0, 0, 0, 1.46313092156811801131018048418, 1.66260061704225131346789765018, 2.79927966384044821933287276777, 3.50437855314430610577113437027, 4.15210514423586779419605478593, 4.23912785895003769931341466511, 5.05832509076758630826783281233, 5.06730892068729817480269668819, 5.88239134962647394570004081477, 6.33632740601128452448267112696, 6.78837948449091974826726938817, 6.82088248415705226338381828771, 7.11217076462862682062123649881, 7.87615370331142558249828812588, 7.901200733731107632500869835234, 8.712145943629755070877300170293, 8.822250236537784233102160049972, 8.883988044396467932812105532638, 9.784012941071971075327506411725, 9.854746844123005947391193913803, 9.987695812572941867857313303660, 10.62572089363730959095947307234, 11.06426070673374402137520530015, 11.31503634546292837979045754117, 11.38396107918053199134896155671

Graph of the ZZ-function along the critical line