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Properties

Label 6-3267e3-1.1-c1e3-0-4
Degree 66
Conductor 3486963516334869635163
Sign −1-1
Analytic cond. 17753.217753.2
Root an. cond. 5.107555.10755
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-s − 4-s − 4·5-s + 4·10-s + 9·13-s + 16-s + 2·17-s − 9·19-s + 4·20-s − 11·23-s + 2·25-s − 9·26-s + 6·29-s + 3·31-s + 2·32-s − 2·34-s − 3·37-s + 9·38-s − 13·41-s − 6·43-s + 11·46-s + 47-s − 12·49-s − 2·50-s − 9·52-s − 15·53-s − 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.26·10-s + 2.49·13-s + 1/4·16-s + 0.485·17-s − 2.06·19-s + 0.894·20-s − 2.29·23-s + 2/5·25-s − 1.76·26-s + 1.11·29-s + 0.538·31-s + 0.353·32-s − 0.342·34-s − 0.493·37-s + 1.45·38-s − 2.03·41-s − 0.914·43-s + 1.62·46-s + 0.145·47-s − 1.71·49-s − 0.282·50-s − 1.24·52-s − 2.06·53-s − 0.787·58-s + ⋯

Functional equation

Λ(s)=((39⋅116)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((39⋅116)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 39⋅1163^{9} \cdot 11^{6}
Sign: −1-1
Analytic conductor: 17753.217753.2
Root analytic conductor: 5.107555.10755
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 39⋅116, ( :1/2,1/2,1/2), −1)(6,\ 3^{9} \cdot 11^{6} ,\ ( \ : 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)
bad3 1 1
11 1 1
good2S4×C2S_4\times C_2 1+T+pT2+3T3+p2T4+p2T5+p3T6 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1+4T+14T2+39T3+14pT4+4p2T5+p3T6 1 + 4 T + 14 T^{2} + 39 T^{3} + 14 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+12T2−T3+12pT4+p3T6 1 + 12 T^{2} - T^{3} + 12 p T^{4} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−9T+3pT2−127T3+3p2T4−9p2T5+p3T6 1 - 9 T + 3 p T^{2} - 127 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−2T+26T2−9T3+26pT4−2p2T5+p3T6 1 - 2 T + 26 T^{2} - 9 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
19C2C_2 (1+3T+pT2)3 ( 1 + 3 T + p T^{2} )^{3}
23S4×C2S_4\times C_2 1+11T+83T2+417T3+83pT4+11p2T5+p3T6 1 + 11 T + 83 T^{2} + 417 T^{3} + 83 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−6T+63T2−276T3+63pT4−6p2T5+p3T6 1 - 6 T + 63 T^{2} - 276 T^{3} + 63 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−3T+27T2+71T3+27pT4−3p2T5+p3T6 1 - 3 T + 27 T^{2} + 71 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+3T+87T2+249T3+87pT4+3p2T5+p3T6 1 + 3 T + 87 T^{2} + 249 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+13T+125T2+1023T3+125pT4+13p2T5+p3T6 1 + 13 T + 125 T^{2} + 1023 T^{3} + 125 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+6T+114T2+417T3+114pT4+6p2T5+p3T6 1 + 6 T + 114 T^{2} + 417 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−T+92T2+27T3+92pT4−p2T5+p3T6 1 - T + 92 T^{2} + 27 T^{3} + 92 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+15T+207T2+1527T3+207pT4+15p2T5+p3T6 1 + 15 T + 207 T^{2} + 1527 T^{3} + 207 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+17T+167T2+1269T3+167pT4+17p2T5+p3T6 1 + 17 T + 167 T^{2} + 1269 T^{3} + 167 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+15T+249T2+1911T3+249pT4+15p2T5+p3T6 1 + 15 T + 249 T^{2} + 1911 T^{3} + 249 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−15T+207T2−1601T3+207pT4−15p2T5+p3T6 1 - 15 T + 207 T^{2} - 1601 T^{3} + 207 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+21T+351T2+3261T3+351pT4+21p2T5+p3T6 1 + 21 T + 351 T^{2} + 3261 T^{3} + 351 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+12T+210T2+1451T3+210pT4+12p2T5+p3T6 1 + 12 T + 210 T^{2} + 1451 T^{3} + 210 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+9T+93T2+523T3+93pT4+9p2T5+p3T6 1 + 9 T + 93 T^{2} + 523 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−15T+141T2−789T3+141pT4−15p2T5+p3T6 1 - 15 T + 141 T^{2} - 789 T^{3} + 141 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1−3T+261T2−525T3+261pT4−3p2T5+p3T6 1 - 3 T + 261 T^{2} - 525 T^{3} + 261 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+9T+162T2+1393T3+162pT4+9p2T5+p3T6 1 + 9 T + 162 T^{2} + 1393 T^{3} + 162 p T^{4} + 9 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

−8.204375577741539805905391496348, −7.918796292283634318282975590120, −7.79218965140110176240108104138, −7.42449227994341860832457707603, −7.16436943282177614203006203197, −6.67310861662956754718582629558, −6.35876439770523109009450693317, −6.31436370108180109139326098809, −6.15367300845571991592322561305, −6.01929781817812614984408401040, −5.57769631650298160676301303105, −5.03726217621722105135733057415, −4.87555150265810344007912857675, −4.51248355399816640552656244334, −4.25311546728134242695827803486, −4.21982522672568026674234858940, −3.66433204197327163612176072218, −3.62049628291257130159528517282, −3.36071661853778916950165734583, −2.94684077108203844122089774163, −2.79830402632950067577343065381, −1.83882784875641966019294920200, −1.78361125904791392605556007052, −1.54945105301474026487199101039, −0.949205263139479937073782570064, 0, 0, 0, 0.949205263139479937073782570064, 1.54945105301474026487199101039, 1.78361125904791392605556007052, 1.83882784875641966019294920200, 2.79830402632950067577343065381, 2.94684077108203844122089774163, 3.36071661853778916950165734583, 3.62049628291257130159528517282, 3.66433204197327163612176072218, 4.21982522672568026674234858940, 4.25311546728134242695827803486, 4.51248355399816640552656244334, 4.87555150265810344007912857675, 5.03726217621722105135733057415, 5.57769631650298160676301303105, 6.01929781817812614984408401040, 6.15367300845571991592322561305, 6.31436370108180109139326098809, 6.35876439770523109009450693317, 6.67310861662956754718582629558, 7.16436943282177614203006203197, 7.42449227994341860832457707603, 7.79218965140110176240108104138, 7.918796292283634318282975590120, 8.204375577741539805905391496348

Graph of the ZZ-function along the critical line