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Properties

Label 6-1632e3-1.1-c1e3-0-0
Degree 66
Conductor 43467079684346707968
Sign 11
Analytic cond. 2213.052213.05
Root an. cond. 3.609923.60992
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 + 3·5-s − 2·7-s + 6·9-s + 11-s + 7·13-s − 9·15-s + 3·17-s − 19-s + 6·21-s − 3·23-s − 2·25-s − 10·27-s + 10·29-s − 10·31-s − 3·33-s − 6·35-s + 8·37-s − 21·39-s + 41-s + 5·43-s + 18·45-s + 4·47-s − 49-s − 9·51-s + 12·53-s + 3·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s + 0.301·11-s + 1.94·13-s − 2.32·15-s + 0.727·17-s − 0.229·19-s + 1.30·21-s − 0.625·23-s − 2/5·25-s − 1.92·27-s + 1.85·29-s − 1.79·31-s − 0.522·33-s − 1.01·35-s + 1.31·37-s − 3.36·39-s + 0.156·41-s + 0.762·43-s + 2.68·45-s + 0.583·47-s − 1/7·49-s − 1.26·51-s + 1.64·53-s + 0.404·55-s + ⋯

Functional equation

Λ(s)=((215⋅33⋅173)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((215⋅33⋅173)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 215⋅33⋅1732^{15} \cdot 3^{3} \cdot 17^{3}
Sign: 11
Analytic conductor: 2213.052213.05
Root analytic conductor: 3.609923.60992
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 215⋅33⋅173, ( :1/2,1/2,1/2), 1)(6,\ 2^{15} \cdot 3^{3} \cdot 17^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.4425297542.442529754
L(12)L(\frac12) ≈\approx 2.4425297542.442529754
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
3C1C_1 (1+T)3 ( 1 + T )^{3}
17C1C_1 (1−T)3 ( 1 - T )^{3}
good5S4×C2S_4\times C_2 1−3T+11T2−26T3+11pT4−3p2T5+p3T6 1 - 3 T + 11 T^{2} - 26 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+2T+5T2+12T3+5pT4+2p2T5+p3T6 1 + 2 T + 5 T^{2} + 12 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−T+17T2−38T3+17pT4−p2T5+p3T6 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−7T+3pT2−178T3+3p2T4−7p2T5+p3T6 1 - 7 T + 3 p T^{2} - 178 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+T+41T2+54T3+41pT4+p2T5+p3T6 1 + T + 41 T^{2} + 54 T^{3} + 41 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+3T+65T2+130T3+65pT4+3p2T5+p3T6 1 + 3 T + 65 T^{2} + 130 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−10T+59T2−236T3+59pT4−10p2T5+p3T6 1 - 10 T + 59 T^{2} - 236 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1+10T+29T2−36T3+29pT4+10p2T5+p3T6 1 + 10 T + 29 T^{2} - 36 T^{3} + 29 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−8T+115T2−560T3+115pT4−8p2T5+p3T6 1 - 8 T + 115 T^{2} - 560 T^{3} + 115 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−T+67T2+90T3+67pT4−p2T5+p3T6 1 - T + 67 T^{2} + 90 T^{3} + 67 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−5T+105T2−446T3+105pT4−5p2T5+p3T6 1 - 5 T + 105 T^{2} - 446 T^{3} + 105 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−4T+77T2−248T3+77pT4−4p2T5+p3T6 1 - 4 T + 77 T^{2} - 248 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−12T+179T2−1208T3+179pT4−12p2T5+p3T6 1 - 12 T + 179 T^{2} - 1208 T^{3} + 179 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−18T+257T2−2156T3+257pT4−18p2T5+p3T6 1 - 18 T + 257 T^{2} - 2156 T^{3} + 257 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−10T+187T2−1132T3+187pT4−10p2T5+p3T6 1 - 10 T + 187 T^{2} - 1132 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−4T+89T2−600T3+89pT4−4p2T5+p3T6 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+4T+157T2+312T3+157pT4+4p2T5+p3T6 1 + 4 T + 157 T^{2} + 312 T^{3} + 157 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−4T+159T2−328T3+159pT4−4p2T5+p3T6 1 - 4 T + 159 T^{2} - 328 T^{3} + 159 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+6T+93T2+516T3+93pT4+6p2T5+p3T6 1 + 6 T + 93 T^{2} + 516 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−20T+137T2−568T3+137pT4−20p2T5+p3T6 1 - 20 T + 137 T^{2} - 568 T^{3} + 137 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+6T+167T2+724T3+167pT4+6p2T5+p3T6 1 + 6 T + 167 T^{2} + 724 T^{3} + 167 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−26T+447T2−5132T3+447pT4−26p2T5+p3T6 1 - 26 T + 447 T^{2} - 5132 T^{3} + 447 p T^{4} - 26 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

−8.594015502636873465158281821512, −7.942431518440154927691131929358, −7.75851013819455600339650013143, −7.47943411447964275621539690898, −7.13947454856216394228161297581, −6.74375000903032064334040761224, −6.60452074660752177535631638824, −6.24873559705433383046865173703, −6.16306778987565442368096537443, −5.91068392849339819467755840037, −5.67257651735996096288572285333, −5.39392241753206658975643861617, −5.24000959711364024903925359218, −4.83080084139464838887104804343, −4.29227166477465897982427676619, −4.09109046053841299230864449990, −3.73872720323261031582607913637, −3.63152726020550958510695039269, −3.15110185021728928535561738674, −2.42140397895564582618677977968, −2.28889447086047333377323055724, −1.88820488620305467050661342958, −1.19821336359186939643948134767, −1.02075818104333479442614380884, −0.54333767801081341089932029220, 0.54333767801081341089932029220, 1.02075818104333479442614380884, 1.19821336359186939643948134767, 1.88820488620305467050661342958, 2.28889447086047333377323055724, 2.42140397895564582618677977968, 3.15110185021728928535561738674, 3.63152726020550958510695039269, 3.73872720323261031582607913637, 4.09109046053841299230864449990, 4.29227166477465897982427676619, 4.83080084139464838887104804343, 5.24000959711364024903925359218, 5.39392241753206658975643861617, 5.67257651735996096288572285333, 5.91068392849339819467755840037, 6.16306778987565442368096537443, 6.24873559705433383046865173703, 6.60452074660752177535631638824, 6.74375000903032064334040761224, 7.13947454856216394228161297581, 7.47943411447964275621539690898, 7.75851013819455600339650013143, 7.942431518440154927691131929358, 8.594015502636873465158281821512

Graph of the ZZ-function along the critical line