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Properties

Label 6-3724e3-1.1-c1e3-0-1
Degree 66
Conductor 5164508742451645087424
Sign −1-1
Analytic cond. 26294.226294.2
Root an. cond. 5.453095.45309
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  − 2·3-s − 3·5-s − 9-s − 5·11-s + 2·13-s + 6·15-s + 10·17-s + 3·19-s − 17·23-s − 25-s + 6·27-s + 2·29-s + 6·31-s + 10·33-s + 14·37-s − 4·39-s + 4·41-s − 17·43-s + 3·45-s − 9·47-s − 20·51-s − 22·53-s + 15·55-s − 6·57-s + 6·59-s + 61-s − 6·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s − 1/3·9-s − 1.50·11-s + 0.554·13-s + 1.54·15-s + 2.42·17-s + 0.688·19-s − 3.54·23-s − 1/5·25-s + 1.15·27-s + 0.371·29-s + 1.07·31-s + 1.74·33-s + 2.30·37-s − 0.640·39-s + 0.624·41-s − 2.59·43-s + 0.447·45-s − 1.31·47-s − 2.80·51-s − 3.02·53-s + 2.02·55-s − 0.794·57-s + 0.781·59-s + 0.128·61-s − 0.744·65-s + ⋯

Functional equation

Λ(s)=((26⋅76⋅193)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅76⋅193)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \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⋅76⋅1932^{6} \cdot 7^{6} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 26294.226294.2
Root analytic conductor: 5.453095.45309
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 26⋅76⋅193, ( :1/2,1/2,1/2), −1)(6,\ 2^{6} \cdot 7^{6} \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
7 1 1
19C1C_1 (1−T)3 ( 1 - T )^{3}
good3S4×C2S_4\times C_2 1+2T+5T2+2pT3+5pT4+2p2T5+p3T6 1 + 2 T + 5 T^{2} + 2 p T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1+3T+2pT2+19T3+2p2T4+3p2T5+p3T6 1 + 3 T + 2 p T^{2} + 19 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+5T+34T2+109T3+34pT4+5p2T5+p3T6 1 + 5 T + 34 T^{2} + 109 T^{3} + 34 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−2T+33T2−54T3+33pT4−2p2T5+p3T6 1 - 2 T + 33 T^{2} - 54 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−10T+67T2−304T3+67pT4−10p2T5+p3T6 1 - 10 T + 67 T^{2} - 304 T^{3} + 67 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+17T+160T2+931T3+160pT4+17p2T5+p3T6 1 + 17 T + 160 T^{2} + 931 T^{3} + 160 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−2T+pT2−134T3+p2T4−2p2T5+p3T6 1 - 2 T + p T^{2} - 134 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−6T+57T2−358T3+57pT4−6p2T5+p3T6 1 - 6 T + 57 T^{2} - 358 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−14T+169T2−30pT3+169pT4−14p2T5+p3T6 1 - 14 T + 169 T^{2} - 30 p T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−4T+51T2−472T3+51pT4−4p2T5+p3T6 1 - 4 T + 51 T^{2} - 472 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+17T+220T2+1611T3+220pT4+17p2T5+p3T6 1 + 17 T + 220 T^{2} + 1611 T^{3} + 220 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+9T+124T2+735T3+124pT4+9p2T5+p3T6 1 + 9 T + 124 T^{2} + 735 T^{3} + 124 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+22T+225T2+1706T3+225pT4+22p2T5+p3T6 1 + 22 T + 225 T^{2} + 1706 T^{3} + 225 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}
59C2C_2 (1−2T+pT2)3 ( 1 - 2 T + p T^{2} )^{3}
61S4×C2S_4\times C_2 1−T+126T2−261T3+126pT4−p2T5+p3T6 1 - T + 126 T^{2} - 261 T^{3} + 126 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−2T+133T2−100T3+133pT4−2p2T5+p3T6 1 - 2 T + 133 T^{2} - 100 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+18T+277T2+2514T3+277pT4+18p2T5+p3T6 1 + 18 T + 277 T^{2} + 2514 T^{3} + 277 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+21T+334T2+3217T3+334pT4+21p2T5+p3T6 1 + 21 T + 334 T^{2} + 3217 T^{3} + 334 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−8T+253T2−1266T3+253pT4−8p2T5+p3T6 1 - 8 T + 253 T^{2} - 1266 T^{3} + 253 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−27T+426T2−4439T3+426pT4−27p2T5+p3T6 1 - 27 T + 426 T^{2} - 4439 T^{3} + 426 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+30T+501T2+5502T3+501pT4+30p2T5+p3T6 1 + 30 T + 501 T^{2} + 5502 T^{3} + 501 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−14T+119T2−1252T3+119pT4−14p2T5+p3T6 1 - 14 T + 119 T^{2} - 1252 T^{3} + 119 p T^{4} - 14 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

−7.917549441894300742215295704536, −7.77771017374071011845873818023, −7.65810837774808098603254207517, −7.34406016692015463907470718202, −6.78725180752673093247316145623, −6.62396745751195080914907822486, −6.33748917171275030514348238141, −6.02714920013909723476670684605, −5.82955778538108564668481829748, −5.79473509243198593522896058360, −5.48053005281760321786826149591, −5.17726112097458194661994816131, −4.91293094674118048737648928705, −4.55255476816345666543820639882, −4.34564161666153776001083134109, −4.16932521480383459554958117862, −3.59756913058653986068066611719, −3.43146768360449209622122638107, −3.34830595301974350672237773885, −2.80244496267558262259992510542, −2.67345445942629267619065552653, −2.25012553492098650550712536626, −1.64139562653932564803890141457, −1.30178853215403316268020547367, −1.02303263923259426795241948509, 0, 0, 0, 1.02303263923259426795241948509, 1.30178853215403316268020547367, 1.64139562653932564803890141457, 2.25012553492098650550712536626, 2.67345445942629267619065552653, 2.80244496267558262259992510542, 3.34830595301974350672237773885, 3.43146768360449209622122638107, 3.59756913058653986068066611719, 4.16932521480383459554958117862, 4.34564161666153776001083134109, 4.55255476816345666543820639882, 4.91293094674118048737648928705, 5.17726112097458194661994816131, 5.48053005281760321786826149591, 5.79473509243198593522896058360, 5.82955778538108564668481829748, 6.02714920013909723476670684605, 6.33748917171275030514348238141, 6.62396745751195080914907822486, 6.78725180752673093247316145623, 7.34406016692015463907470718202, 7.65810837774808098603254207517, 7.77771017374071011845873818023, 7.917549441894300742215295704536

Graph of the ZZ-function along the critical line