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Properties

Label 6-7920e3-1.1-c1e3-0-0
Degree 66
Conductor 496793088000496793088000
Sign 11
Analytic cond. 252933.252933.
Root an. cond. 7.952457.95245
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 3·11-s + 4·13-s − 6·17-s − 2·19-s − 2·23-s + 6·25-s − 6·29-s + 4·31-s + 6·37-s − 6·41-s + 2·43-s + 14·47-s − 7·49-s − 6·53-s − 9·55-s + 4·59-s − 12·65-s + 12·67-s + 10·71-s − 6·73-s + 4·79-s + 4·83-s + 18·85-s − 12·89-s + 6·95-s + 2·97-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.904·11-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.417·23-s + 6/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.304·43-s + 2.04·47-s − 49-s − 0.824·53-s − 1.21·55-s + 0.520·59-s − 1.48·65-s + 1.46·67-s + 1.18·71-s − 0.702·73-s + 0.450·79-s + 0.439·83-s + 1.95·85-s − 1.27·89-s + 0.615·95-s + 0.203·97-s + ⋯

Functional equation

Λ(s)=((212⋅36⋅53⋅113)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅36⋅53⋅113)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 212⋅36⋅53⋅1132^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}
Sign: 11
Analytic conductor: 252933.252933.
Root analytic conductor: 7.952457.95245
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 212⋅36⋅53⋅113, ( :1/2,1/2,1/2), 1)(6,\ 2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.0450120353.045012035
L(12)L(\frac12) ≈\approx 3.0450120353.045012035
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
5C1C_1 (1+T)3 ( 1 + T )^{3}
11C1C_1 (1−T)3 ( 1 - T )^{3}
good7S4×C2S_4\times C_2 1+pT2−16T3+p2T4+p3T6 1 + p T^{2} - 16 T^{3} + p^{2} T^{4} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−4T+pT2−8T3+p2T4−4p2T5+p3T6 1 - 4 T + p T^{2} - 8 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+6T+49T2+168T3+49pT4+6p2T5+p3T6 1 + 6 T + 49 T^{2} + 168 T^{3} + 49 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+2T+33T2+60T3+33pT4+2p2T5+p3T6 1 + 2 T + 33 T^{2} + 60 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+2T+45T2+76T3+45pT4+2p2T5+p3T6 1 + 2 T + 45 T^{2} + 76 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
29C2C_2 (1+2T+pT2)3 ( 1 + 2 T + p T^{2} )^{3}
31S4×C2S_4\times C_2 1−4T+73T2−216T3+73pT4−4p2T5+p3T6 1 - 4 T + 73 T^{2} - 216 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
37C2C_2 (1−2T+pT2)3 ( 1 - 2 T + p T^{2} )^{3}
41C2C_2 (1+2T+pT2)3 ( 1 + 2 T + p T^{2} )^{3}
43S4×C2S_4\times C_2 1−2T+83T2−16T3+83pT4−2p2T5+p3T6 1 - 2 T + 83 T^{2} - 16 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−14T+181T2−1300T3+181pT4−14p2T5+p3T6 1 - 14 T + 181 T^{2} - 1300 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+6T+115T2+404T3+115pT4+6p2T5+p3T6 1 + 6 T + 115 T^{2} + 404 T^{3} + 115 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−4T+157T2−440T3+157pT4−4p2T5+p3T6 1 - 4 T + 157 T^{2} - 440 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−45T2−16T3−45pT4+p3T6 1 - 45 T^{2} - 16 T^{3} - 45 p T^{4} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−12T+193T2−1576T3+193pT4−12p2T5+p3T6 1 - 12 T + 193 T^{2} - 1576 T^{3} + 193 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−10T+121T2−1492T3+121pT4−10p2T5+p3T6 1 - 10 T + 121 T^{2} - 1492 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+6T+105T2+200T3+105pT4+6p2T5+p3T6 1 + 6 T + 105 T^{2} + 200 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−4T+117T2−920T3+117pT4−4p2T5+p3T6 1 - 4 T + 117 T^{2} - 920 T^{3} + 117 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−4T−21T2+1168T3−21pT4−4p2T5+p3T6 1 - 4 T - 21 T^{2} + 1168 T^{3} - 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+12T−17T2−1320T3−17pT4+12p2T5+p3T6 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−2T+103T2−1260T3+103pT4−2p2T5+p3T6 1 - 2 T + 103 T^{2} - 1260 T^{3} + 103 p T^{4} - 2 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

−7.08588538246036496353085511327, −6.57565914071588866255000517095, −6.46952288873415835745020589509, −6.42923900041193145096099218526, −6.00607031873828036620076040268, −5.83294363793922668496261532781, −5.65182422483677650554724055030, −5.09463510357571939565624763040, −4.94769682880040317224161619617, −4.92186902152704414789994707432, −4.28713143687582927553303061746, −4.21440329507926158668823124887, −4.09962227094541030688644284897, −3.82555965982813357286118015104, −3.67989048443947794201973746108, −3.31197152145623431429711060555, −3.02496804240663165493719755301, −2.62448910976994393767144653487, −2.53773993258718925036281094637, −1.99230321558861440120664656278, −1.69169927610135160158495235336, −1.59549443629513187607759390112, −0.927976871389569294018248088191, −0.59546545722128343648614875853, −0.40568118235162382630963376134, 0.40568118235162382630963376134, 0.59546545722128343648614875853, 0.927976871389569294018248088191, 1.59549443629513187607759390112, 1.69169927610135160158495235336, 1.99230321558861440120664656278, 2.53773993258718925036281094637, 2.62448910976994393767144653487, 3.02496804240663165493719755301, 3.31197152145623431429711060555, 3.67989048443947794201973746108, 3.82555965982813357286118015104, 4.09962227094541030688644284897, 4.21440329507926158668823124887, 4.28713143687582927553303061746, 4.92186902152704414789994707432, 4.94769682880040317224161619617, 5.09463510357571939565624763040, 5.65182422483677650554724055030, 5.83294363793922668496261532781, 6.00607031873828036620076040268, 6.42923900041193145096099218526, 6.46952288873415835745020589509, 6.57565914071588866255000517095, 7.08588538246036496353085511327

Graph of the ZZ-function along the critical line