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Properties

Label 6-825e3-1.1-c3e3-0-1
Degree 66
Conductor 561515625561515625
Sign 11
Analytic cond. 115334.115334.
Root an. cond. 6.976866.97686
Motivic weight 33
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  + 4·2-s + 9·3-s + 7·4-s + 36·6-s + 4·7-s + 12·8-s + 54·9-s + 33·11-s + 63·12-s + 16·14-s − 7·16-s + 218·17-s + 216·18-s + 146·19-s + 36·21-s + 132·22-s + 200·23-s + 108·24-s + 270·27-s + 28·28-s + 68·29-s − 68·31-s − 28·32-s + 297·33-s + 872·34-s + 378·36-s + 390·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 7/8·4-s + 2.44·6-s + 0.215·7-s + 0.530·8-s + 2·9-s + 0.904·11-s + 1.51·12-s + 0.305·14-s − 0.109·16-s + 3.11·17-s + 2.82·18-s + 1.76·19-s + 0.374·21-s + 1.27·22-s + 1.81·23-s + 0.918·24-s + 1.92·27-s + 0.188·28-s + 0.435·29-s − 0.393·31-s − 0.154·32-s + 1.56·33-s + 4.39·34-s + 7/4·36-s + 1.73·37-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅113)s/2 ΓC(s)3 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((33⋅56⋅113)s/2 ΓC(s+3/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅1133^{3} \cdot 5^{6} \cdot 11^{3}
Sign: 11
Analytic conductor: 115334.115334.
Root analytic conductor: 6.976866.97686
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅113, ( :3/2,3/2,3/2), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 45.3451368245.34513682
L(12)L(\frac12) ≈\approx 45.3451368245.34513682
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}
5 1 1
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

−8.824538265957090862984442657588, −8.274669889890678787790145893236, −8.141253089791682904017279604648, −7.70231136854552248755973304148, −7.60649173355699630157614715211, −7.21641676367675934544609925841, −7.19538017647700608216589101276, −6.74517118727960106577098322585, −6.14842024987554433132806232754, −6.08708190831412417499210704726, −5.48565476795241888890576898314, −5.45186028111385934401011142552, −4.95279547076424105681938167816, −4.67588187526523506948114370755, −4.31951304257228794079168632437, −4.12390094827828533918595865420, −3.51096882456687860401636107810, −3.35697234964986496513408637015, −3.17039257360102443364282472464, −2.67814348647968432391518070514, −2.66408071374131149618080691913, −1.65470999538928771380172920643, −1.50455861492090021604115178052, −0.978278863960150269743198770832, −0.812636248968871468148556731499, 0.812636248968871468148556731499, 0.978278863960150269743198770832, 1.50455861492090021604115178052, 1.65470999538928771380172920643, 2.66408071374131149618080691913, 2.67814348647968432391518070514, 3.17039257360102443364282472464, 3.35697234964986496513408637015, 3.51096882456687860401636107810, 4.12390094827828533918595865420, 4.31951304257228794079168632437, 4.67588187526523506948114370755, 4.95279547076424105681938167816, 5.45186028111385934401011142552, 5.48565476795241888890576898314, 6.08708190831412417499210704726, 6.14842024987554433132806232754, 6.74517118727960106577098322585, 7.19538017647700608216589101276, 7.21641676367675934544609925841, 7.60649173355699630157614715211, 7.70231136854552248755973304148, 8.141253089791682904017279604648, 8.274669889890678787790145893236, 8.824538265957090862984442657588

Graph of the ZZ-function along the critical line