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Properties

Label 4-2496e2-1.1-c3e2-0-12
Degree 44
Conductor 62300166230016
Sign 11
Analytic cond. 21688.021688.0
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 6·5-s + 10·7-s + 27·9-s − 16·11-s − 26·13-s + 36·15-s + 4·17-s − 70·19-s − 60·21-s + 128·23-s − 110·25-s − 108·27-s + 80·29-s + 250·31-s + 96·33-s − 60·35-s + 152·37-s + 156·39-s − 146·41-s − 504·43-s − 162·45-s + 524·47-s − 498·49-s − 24·51-s + 52·53-s + 96·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.536·5-s + 0.539·7-s + 9-s − 0.438·11-s − 0.554·13-s + 0.619·15-s + 0.0570·17-s − 0.845·19-s − 0.623·21-s + 1.16·23-s − 0.879·25-s − 0.769·27-s + 0.512·29-s + 1.44·31-s + 0.506·33-s − 0.289·35-s + 0.675·37-s + 0.640·39-s − 0.556·41-s − 1.78·43-s − 0.536·45-s + 1.62·47-s − 1.45·49-s − 0.0658·51-s + 0.134·53-s + 0.235·55-s + ⋯

Functional equation

Λ(s)=(6230016s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(6230016s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 62300166230016    =    212⋅32⋅1322^{12} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 21688.021688.0
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 6230016, ( :3/2,3/2), 1)(4,\ 6230016,\ (\ :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)
bad2 1 1
3C1C_1 (1+pT)2 ( 1 + p T )^{2}
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1+6T+146T2+6p3T3+p6T4 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−10T+598T2−10p3T3+p6T4 1 - 10 T + 598 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+16T+918T2+16p3T3+p6T4 1 + 16 T + 918 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1−2T+p3T2)2 ( 1 - 2 T + p^{3} T^{2} )^{2}
19D4D_{4} 1+70T+12118T2+70p3T3+p6T4 1 + 70 T + 12118 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4}
23C2C_2 (1−64T+p3T2)2 ( 1 - 64 T + p^{3} T^{2} )^{2}
29D4D_{4} 1−80T+46310T2−80p3T3+p6T4 1 - 80 T + 46310 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−250T+49782T2−250p3T3+p6T4 1 - 250 T + 49782 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−152T+70470T2−152p3T3+p6T4 1 - 152 T + 70470 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+146T+137634T2+146p3T3+p6T4 1 + 146 T + 137634 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+504T+215286T2+504p3T3+p6T4 1 + 504 T + 215286 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−524T+272222T2−524p3T3+p6T4 1 - 524 T + 272222 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−52T+291198T2−52p3T3+p6T4 1 - 52 T + 291198 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+164T+406182T2+164p3T3+p6T4 1 + 164 T + 406182 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−304T+237958T2−304p3T3+p6T4 1 - 304 T + 237958 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+914T+804838T2+914p3T3+p6T4 1 + 914 T + 804838 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4}
71C22C_2^2 1+455470T2+p6T4 1 + 455470 T^{2} + p^{6} T^{4}
73D4D_{4} 1+456T+825950T2+456p3T3+p6T4 1 + 456 T + 825950 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−824T+1009374T2−824p3T3+p6T4 1 - 824 T + 1009374 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−828T+1032470T2−828p3T3+p6T4 1 - 828 T + 1032470 T^{2} - 828 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−826T+1571354T2−826p3T3+p6T4 1 - 826 T + 1571354 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−552T+219630T2−552p3T3+p6T4 1 - 552 T + 219630 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.274090572441093308961868206413, −7.972556742047539197724692053360, −7.60692503212540092891439426325, −7.22744247796560373984866853908, −6.81036324951670558010687840380, −6.41211384218241928822686503231, −6.10500889564802934571012423275, −5.66825137743847929935455953648, −5.03646704181157927918863703045, −4.94699665447484958439676613116, −4.39812110862154928310069838310, −4.32279236506706821650487693490, −3.33861678365140594515102483672, −3.28707266826865341493779101878, −2.30859194913056508998304192021, −2.14336848176103515447395174572, −1.22193273094220616114550804154, −0.949913636232847777923886681029, 0, 0, 0.949913636232847777923886681029, 1.22193273094220616114550804154, 2.14336848176103515447395174572, 2.30859194913056508998304192021, 3.28707266826865341493779101878, 3.33861678365140594515102483672, 4.32279236506706821650487693490, 4.39812110862154928310069838310, 4.94699665447484958439676613116, 5.03646704181157927918863703045, 5.66825137743847929935455953648, 6.10500889564802934571012423275, 6.41211384218241928822686503231, 6.81036324951670558010687840380, 7.22744247796560373984866853908, 7.60692503212540092891439426325, 7.972556742047539197724692053360, 8.274090572441093308961868206413

Graph of the ZZ-function along the critical line