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Properties

Label 6-693e3-1.1-c1e3-0-0
Degree 66
Conductor 332812557332812557
Sign 11
Analytic cond. 169.445169.445
Root an. cond. 2.352362.35236
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  − 3·7-s − 8-s − 3·11-s + 12·19-s + 6·23-s − 12·29-s − 6·31-s − 6·41-s + 6·43-s + 24·47-s + 6·49-s + 3·56-s + 24·59-s + 18·61-s − 7·64-s + 12·67-s − 12·71-s + 24·73-s + 9·77-s + 12·79-s + 18·83-s + 3·88-s − 18·89-s + 24·97-s + 12·101-s + 12·107-s − 12·109-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.353·8-s − 0.904·11-s + 2.75·19-s + 1.25·23-s − 2.22·29-s − 1.07·31-s − 0.937·41-s + 0.914·43-s + 3.50·47-s + 6/7·49-s + 0.400·56-s + 3.12·59-s + 2.30·61-s − 7/8·64-s + 1.46·67-s − 1.42·71-s + 2.80·73-s + 1.02·77-s + 1.35·79-s + 1.97·83-s + 0.319·88-s − 1.90·89-s + 2.43·97-s + 1.19·101-s + 1.16·107-s − 1.14·109-s + ⋯

Functional equation

Λ(s)=((36⋅73⋅113)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((36⋅73⋅113)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{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: 36⋅73⋅1133^{6} \cdot 7^{3} \cdot 11^{3}
Sign: 11
Analytic conductor: 169.445169.445
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 36⋅73⋅113, ( :1/2,1/2,1/2), 1)(6,\ 3^{6} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.0663563482.066356348
L(12)L(\frac12) ≈\approx 2.0663563482.066356348
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
7C1C_1 (1+T)3 ( 1 + T )^{3}
11C1C_1 (1+T)3 ( 1 + T )^{3}
good2D6D_{6} 1+T3+p3T6 1 + T^{3} + p^{3} T^{6}
5D6D_{6} 1−2T3+p3T6 1 - 2 T^{3} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+24T2+2T3+24pT4+p3T6 1 + 24 T^{2} + 2 T^{3} + 24 p T^{4} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+27T2−8T3+27pT4+p3T6 1 + 27 T^{2} - 8 T^{3} + 27 p T^{4} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−12T+84T2−420T3+84pT4−12p2T5+p3T6 1 - 12 T + 84 T^{2} - 420 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−6T+57T2−244T3+57pT4−6p2T5+p3T6 1 - 6 T + 57 T^{2} - 244 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + 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+6T+57T2+404T3+57pT4+6p2T5+p3T6 1 + 6 T + 57 T^{2} + 404 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+36T2−246T3+36pT4+p3T6 1 + 36 T^{2} - 246 T^{3} + 36 p T^{4} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+6T+51T2+460T3+51pT4+6p2T5+p3T6 1 + 6 T + 51 T^{2} + 460 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−6T+117T2−468T3+117pT4−6p2T5+p3T6 1 - 6 T + 117 T^{2} - 468 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−24T+312T2−2584T3+312pT4−24p2T5+p3T6 1 - 24 T + 312 T^{2} - 2584 T^{3} + 312 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+111T2+120T3+111pT4+p3T6 1 + 111 T^{2} + 120 T^{3} + 111 p T^{4} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−24T+264T2−2116T3+264pT4−24p2T5+p3T6 1 - 24 T + 264 T^{2} - 2116 T^{3} + 264 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6}
61C2C_2 (1−6T+pT2)3 ( 1 - 6 T + p T^{2} )^{3}
67S4×C2S_4\times C_2 1−12T+228T2−1604T3+228pT4−12p2T5+p3T6 1 - 12 T + 228 T^{2} - 1604 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+12T+165T2+1320T3+165pT4+12p2T5+p3T6 1 + 12 T + 165 T^{2} + 1320 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−24T+396T2−3898T3+396pT4−24p2T5+p3T6 1 - 24 T + 396 T^{2} - 3898 T^{3} + 396 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−12T+189T2−1640T3+189pT4−12p2T5+p3T6 1 - 12 T + 189 T^{2} - 1640 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−18T+309T2−3036T3+309pT4−18p2T5+p3T6 1 - 18 T + 309 T^{2} - 3036 T^{3} + 309 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+18T+183T2+1308T3+183pT4+18p2T5+p3T6 1 + 18 T + 183 T^{2} + 1308 T^{3} + 183 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−24T+435T2−4664T3+435pT4−24p2T5+p3T6 1 - 24 T + 435 T^{2} - 4664 T^{3} + 435 p T^{4} - 24 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

−9.350735673943346928605407243924, −9.009891951703012985039789877855, −8.791199579904368345814017220001, −8.751931211758680633962335380238, −8.016988979946331964722263858808, −7.79978326617433976009661469734, −7.29271467714987259448403057269, −7.27945009656910062558633431494, −7.26462857320688202185871362338, −6.61026356290737500466431849547, −6.42103710521066609766729256108, −5.80565782620660755873390959829, −5.57451343138068983183318816028, −5.30898096073083919843041334668, −5.30200701719432773777941487258, −4.84980053666902816021388289595, −4.06983760073111197019964568070, −3.76758633823650279172768131407, −3.50835113085756827866004396710, −3.32704169591219710646693010646, −2.71824404546642850855812380802, −2.29477518003028835573674020964, −2.07452474876015681857580409472, −0.849121879007399067292849754039, −0.74665830070033025775674089900, 0.74665830070033025775674089900, 0.849121879007399067292849754039, 2.07452474876015681857580409472, 2.29477518003028835573674020964, 2.71824404546642850855812380802, 3.32704169591219710646693010646, 3.50835113085756827866004396710, 3.76758633823650279172768131407, 4.06983760073111197019964568070, 4.84980053666902816021388289595, 5.30200701719432773777941487258, 5.30898096073083919843041334668, 5.57451343138068983183318816028, 5.80565782620660755873390959829, 6.42103710521066609766729256108, 6.61026356290737500466431849547, 7.26462857320688202185871362338, 7.27945009656910062558633431494, 7.29271467714987259448403057269, 7.79978326617433976009661469734, 8.016988979946331964722263858808, 8.751931211758680633962335380238, 8.791199579904368345814017220001, 9.009891951703012985039789877855, 9.350735673943346928605407243924

Graph of the ZZ-function along the critical line