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Properties

Label 6-825e3-1.1-c3e3-0-4
Degree 66
Conductor 561515625561515625
Sign −1-1
Analytic cond. 115334.115334.
Root an. cond. 6.976866.97686
Motivic weight 33
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 + 9·3-s − 14·4-s − 9·6-s + 16·7-s + 18·8-s + 54·9-s − 33·11-s − 126·12-s + 42·13-s − 16·14-s + 83·16-s + 34·17-s − 54·18-s − 280·19-s + 144·21-s + 33·22-s + 112·23-s + 162·24-s − 42·26-s + 270·27-s − 224·28-s − 290·29-s − 392·31-s − 143·32-s − 297·33-s − 34·34-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.73·3-s − 7/4·4-s − 0.612·6-s + 0.863·7-s + 0.795·8-s + 2·9-s − 0.904·11-s − 3.03·12-s + 0.896·13-s − 0.305·14-s + 1.29·16-s + 0.485·17-s − 0.707·18-s − 3.38·19-s + 1.49·21-s + 0.319·22-s + 1.01·23-s + 1.37·24-s − 0.316·26-s + 1.92·27-s − 1.51·28-s − 1.85·29-s − 2.27·31-s − 0.789·32-s − 1.56·33-s − 0.171·34-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅113)s/2 ΓC(s)3 L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}
Λ(s)=((33⋅56⋅113)s/2 ΓC(s+3/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅1133^{3} \cdot 5^{6} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 115334.115334.
Root analytic conductor: 6.976866.97686
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 33⋅56⋅113, ( :3/2,3/2,3/2), −1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3C1C_1 (1−pT)3 ( 1 - p T )^{3}
5 1 1
11C1C_1 (1+pT)3 ( 1 + p T )^{3}
good2S4×C2S_4\times C_2 1+T+15T2+11T3+15p3T4+p6T5+p9T6 1 + T + 15 T^{2} + 11 T^{3} + 15 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6}
7S4×C2S_4\times C_2 1−16T+277T2−5168T3+277p3T4−16p6T5+p9T6 1 - 16 T + 277 T^{2} - 5168 T^{3} + 277 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6}
13S4×C2S_4\times C_2 1−42T+1363T2−47132T3+1363p3T4−42p6T5+p9T6 1 - 42 T + 1363 T^{2} - 47132 T^{3} + 1363 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6}
17S4×C2S_4\times C_2 1−2pT+10079T2−155020T3+10079p3T4−2p7T5+p9T6 1 - 2 p T + 10079 T^{2} - 155020 T^{3} + 10079 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6}
19S4×C2S_4\times C_2 1+280T+39201T2+3743984T3+39201p3T4+280p6T5+p9T6 1 + 280 T + 39201 T^{2} + 3743984 T^{3} + 39201 p^{3} T^{4} + 280 p^{6} T^{5} + p^{9} T^{6}
23S4×C2S_4\times C_2 1−112T+19477T2−809120T3+19477p3T4−112p6T5+p9T6 1 - 112 T + 19477 T^{2} - 809120 T^{3} + 19477 p^{3} T^{4} - 112 p^{6} T^{5} + p^{9} T^{6}
29S4×C2S_4\times C_2 1+10pT+46667T2+4894124T3+46667p3T4+10p7T5+p9T6 1 + 10 p T + 46667 T^{2} + 4894124 T^{3} + 46667 p^{3} T^{4} + 10 p^{7} T^{5} + p^{9} T^{6}
31S4×C2S_4\times C_2 1+392T+121533T2+23039344T3+121533p3T4+392p6T5+p9T6 1 + 392 T + 121533 T^{2} + 23039344 T^{3} + 121533 p^{3} T^{4} + 392 p^{6} T^{5} + p^{9} T^{6}
37S4×C2S_4\times C_2 1+570T+212899T2+56704796T3+212899p3T4+570p6T5+p9T6 1 + 570 T + 212899 T^{2} + 56704796 T^{3} + 212899 p^{3} T^{4} + 570 p^{6} T^{5} + p^{9} T^{6}
41S4×C2S_4\times C_2 1+662T+150855T2+22689620T3+150855p3T4+662p6T5+p9T6 1 + 662 T + 150855 T^{2} + 22689620 T^{3} + 150855 p^{3} T^{4} + 662 p^{6} T^{5} + p^{9} T^{6}
43S4×C2S_4\times C_2 1−68T+11145T2+9678104T3+11145p3T4−68p6T5+p9T6 1 - 68 T + 11145 T^{2} + 9678104 T^{3} + 11145 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6}
47S4×C2S_4\times C_2 1+264T+146893T2+68940144T3+146893p3T4+264p6T5+p9T6 1 + 264 T + 146893 T^{2} + 68940144 T^{3} + 146893 p^{3} T^{4} + 264 p^{6} T^{5} + p^{9} T^{6}
53S4×C2S_4\times C_2 1−94T+388819T2−25584884T3+388819p3T4−94p6T5+p9T6 1 - 94 T + 388819 T^{2} - 25584884 T^{3} + 388819 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6}
59S4×C2S_4\times C_2 1+612T+191353T2+89255576T3+191353p3T4+612p6T5+p9T6 1 + 612 T + 191353 T^{2} + 89255576 T^{3} + 191353 p^{3} T^{4} + 612 p^{6} T^{5} + p^{9} T^{6}
61S4×C2S_4\times C_2 1+582T+700987T2+242850884T3+700987p3T4+582p6T5+p9T6 1 + 582 T + 700987 T^{2} + 242850884 T^{3} + 700987 p^{3} T^{4} + 582 p^{6} T^{5} + p^{9} T^{6}
67S4×C2S_4\times C_2 1+940T+512801T2+170936200T3+512801p3T4+940p6T5+p9T6 1 + 940 T + 512801 T^{2} + 170936200 T^{3} + 512801 p^{3} T^{4} + 940 p^{6} T^{5} + p^{9} T^{6}
71S4×C2S_4\times C_2 1+1616T+1744933T2+1196967136T3+1744933p3T4+1616p6T5+p9T6 1 + 1616 T + 1744933 T^{2} + 1196967136 T^{3} + 1744933 p^{3} T^{4} + 1616 p^{6} T^{5} + p^{9} T^{6}
73S4×C2S_4\times C_2 1+738T+1335919T2+586235356T3+1335919p3T4+738p6T5+p9T6 1 + 738 T + 1335919 T^{2} + 586235356 T^{3} + 1335919 p^{3} T^{4} + 738 p^{6} T^{5} + p^{9} T^{6}
79S4×C2S_4\times C_2 1−124T+1293245T2−95402344T3+1293245p3T4−124p6T5+p9T6 1 - 124 T + 1293245 T^{2} - 95402344 T^{3} + 1293245 p^{3} T^{4} - 124 p^{6} T^{5} + p^{9} T^{6}
83S4×C2S_4\times C_2 1−1232T+2008521T2−1393226768T3+2008521p3T4−1232p6T5+p9T6 1 - 1232 T + 2008521 T^{2} - 1393226768 T^{3} + 2008521 p^{3} T^{4} - 1232 p^{6} T^{5} + p^{9} T^{6}
89S4×C2S_4\times C_2 1−838T+1110375T2−349581812T3+1110375p3T4−838p6T5+p9T6 1 - 838 T + 1110375 T^{2} - 349581812 T^{3} + 1110375 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6}
97S4×C2S_4\times C_2 1−90T+441679T2+759889556T3+441679p3T4−90p6T5+p9T6 1 - 90 T + 441679 T^{2} + 759889556 T^{3} + 441679 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} 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

−8.975257984627867981597137714799, −8.756886792663463848002055456392, −8.664874130697926529337598270123, −8.644008272762849038412990189364, −8.062074708593717934400554198884, −7.83612037750024537156086712475, −7.55847616218115568494835181138, −7.44909690622839603019137495634, −6.87360478448751635380286870537, −6.72604814050844483706955404452, −6.04175695660354099526665503266, −6.01966312603529477032849909912, −5.32961516232190930103421756989, −5.10611389301653612594103417886, −4.76564008235122752127719055300, −4.68472080083570909399073377723, −4.07271715208151433876052641457, −3.85092018567788053733613541436, −3.69672774747570427612935804381, −3.08737524024092746834732223996, −3.06091685741671221142367814818, −2.12551233059698608939943653504, −1.90538860750957338173778733004, −1.61551328094500872726920373329, −1.34100164728806702134706110119, 0, 0, 0, 1.34100164728806702134706110119, 1.61551328094500872726920373329, 1.90538860750957338173778733004, 2.12551233059698608939943653504, 3.06091685741671221142367814818, 3.08737524024092746834732223996, 3.69672774747570427612935804381, 3.85092018567788053733613541436, 4.07271715208151433876052641457, 4.68472080083570909399073377723, 4.76564008235122752127719055300, 5.10611389301653612594103417886, 5.32961516232190930103421756989, 6.01966312603529477032849909912, 6.04175695660354099526665503266, 6.72604814050844483706955404452, 6.87360478448751635380286870537, 7.44909690622839603019137495634, 7.55847616218115568494835181138, 7.83612037750024537156086712475, 8.062074708593717934400554198884, 8.644008272762849038412990189364, 8.664874130697926529337598270123, 8.756886792663463848002055456392, 8.975257984627867981597137714799

Graph of the ZZ-function along the critical line