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Properties

Label 6-8112e3-1.1-c1e3-0-3
Degree 66
Conductor 533806460928533806460928
Sign 11
Analytic cond. 271778.271778.
Root an. cond. 8.048268.04826
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  + 3·3-s + 3·7-s + 6·9-s + 6·19-s + 9·21-s + 10·27-s − 12·29-s + 3·31-s − 6·37-s + 21·43-s − 12·47-s + 6·49-s − 6·53-s + 18·57-s + 6·59-s + 3·61-s + 18·63-s + 27·67-s + 12·71-s + 9·73-s − 9·79-s + 15·81-s − 18·83-s − 36·87-s − 24·89-s + 9·93-s − 21·97-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.13·7-s + 2·9-s + 1.37·19-s + 1.96·21-s + 1.92·27-s − 2.22·29-s + 0.538·31-s − 0.986·37-s + 3.20·43-s − 1.75·47-s + 6/7·49-s − 0.824·53-s + 2.38·57-s + 0.781·59-s + 0.384·61-s + 2.26·63-s + 3.29·67-s + 1.42·71-s + 1.05·73-s − 1.01·79-s + 5/3·81-s − 1.97·83-s − 3.85·87-s − 2.54·89-s + 0.933·93-s − 2.13·97-s + ⋯

Functional equation

Λ(s)=((212⋅33⋅136)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅33⋅136)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 212⋅33⋅1362^{12} \cdot 3^{3} \cdot 13^{6}
Sign: 11
Analytic conductor: 271778.271778.
Root analytic conductor: 8.048268.04826
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 212⋅33⋅136, ( :1/2,1/2,1/2), 1)(6,\ 2^{12} \cdot 3^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 14.6256605314.62566053
L(12)L(\frac12) ≈\approx 14.6256605314.62566053
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}
13 1 1
good5D6D_{6} 1−2T3+p3T6 1 - 2 T^{3} + p^{3} T^{6}
7S4×C2S_4\times C_2 1−3T+3T2−6T3+3pT4−3p2T5+p3T6 1 - 3 T + 3 T^{2} - 6 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+9T2+8T3+9pT4+p3T6 1 + 9 T^{2} + 8 T^{3} + 9 p T^{4} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+30T2+16T3+30pT4+p3T6 1 + 30 T^{2} + 16 T^{3} + 30 p T^{4} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−6T+21T2−20T3+21pT4−6p2T5+p3T6 1 - 6 T + 21 T^{2} - 20 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+45T2−8T3+45pT4+p3T6 1 + 45 T^{2} - 8 T^{3} + 45 p T^{4} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+12T+120T2+702T3+120pT4+12p2T5+p3T6 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−3T+81T2−170T3+81pT4−3p2T5+p3T6 1 - 3 T + 81 T^{2} - 170 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+6T+102T2+426T3+102pT4+6p2T5+p3T6 1 + 6 T + 102 T^{2} + 426 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+66T2−156T3+66pT4+p3T6 1 + 66 T^{2} - 156 T^{3} + 66 p T^{4} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−21T+255T2−2018T3+255pT4−21p2T5+p3T6 1 - 21 T + 255 T^{2} - 2018 T^{3} + 255 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+12T+165T2+1104T3+165pT4+12p2T5+p3T6 1 + 12 T + 165 T^{2} + 1104 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+6T+36T2+428T3+36pT4+6p2T5+p3T6 1 + 6 T + 36 T^{2} + 428 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−6T+105T2−676T3+105pT4−6p2T5+p3T6 1 - 6 T + 105 T^{2} - 676 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−3T+138T2−199T3+138pT4−3p2T5+p3T6 1 - 3 T + 138 T^{2} - 199 T^{3} + 138 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−27T+423T2−4142T3+423pT4−27p2T5+p3T6 1 - 27 T + 423 T^{2} - 4142 T^{3} + 423 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−12T+237T2−1664T3+237pT4−12p2T5+p3T6 1 - 12 T + 237 T^{2} - 1664 T^{3} + 237 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−9T+186T2−1145T3+186pT4−9p2T5+p3T6 1 - 9 T + 186 T^{2} - 1145 T^{3} + 186 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+9T+117T2+1438T3+117pT4+9p2T5+p3T6 1 + 9 T + 117 T^{2} + 1438 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+18T+237T2+1996T3+237pT4+18p2T5+p3T6 1 + 18 T + 237 T^{2} + 1996 T^{3} + 237 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+24T+399T2+4320T3+399pT4+24p2T5+p3T6 1 + 24 T + 399 T^{2} + 4320 T^{3} + 399 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+21T+423T2+4310T3+423pT4+21p2T5+p3T6 1 + 21 T + 423 T^{2} + 4310 T^{3} + 423 p T^{4} + 21 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

−7.12404639569665187877935060491, −6.72852343416445593432092816184, −6.66083898543128818275623930313, −6.33474474586811010497403058514, −5.77814599739960985306150567832, −5.62722860431681652715986995653, −5.56150949518056302223046549025, −5.17577021097293324623786237849, −5.17082425503951890388431932340, −4.72850975996421025499019549823, −4.27881130825608904109717942493, −4.23153084687601711150844864763, −4.11237685736333142575421111520, −3.69398370866153121224039493738, −3.47363584932659434163518151631, −3.29596905097072717731036906283, −2.84163844020425474232110561602, −2.70764088629755819356285101718, −2.47059240046677926020914839401, −1.99830498035532661553437333961, −1.73821565662767161778269891108, −1.72420577059930845981412376274, −1.22502617332601423812403138027, −0.77662766392430302970864618650, −0.50580468635573978042579600483, 0.50580468635573978042579600483, 0.77662766392430302970864618650, 1.22502617332601423812403138027, 1.72420577059930845981412376274, 1.73821565662767161778269891108, 1.99830498035532661553437333961, 2.47059240046677926020914839401, 2.70764088629755819356285101718, 2.84163844020425474232110561602, 3.29596905097072717731036906283, 3.47363584932659434163518151631, 3.69398370866153121224039493738, 4.11237685736333142575421111520, 4.23153084687601711150844864763, 4.27881130825608904109717942493, 4.72850975996421025499019549823, 5.17082425503951890388431932340, 5.17577021097293324623786237849, 5.56150949518056302223046549025, 5.62722860431681652715986995653, 5.77814599739960985306150567832, 6.33474474586811010497403058514, 6.66083898543128818275623930313, 6.72852343416445593432092816184, 7.12404639569665187877935060491

Graph of the ZZ-function along the critical line