↑

Properties

Label 4-11e4-1.1-c3e2-0-2
Degree 44
Conductor 1464114641
Sign 11
Analytic cond. 50.968650.9686
Root an. cond. 2.671932.67193
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 10·4-s + 2·5-s + 4·6-s − 20·7-s + 32·8-s − 3·9-s − 4·10-s + 20·12-s − 80·13-s + 40·14-s − 4·15-s + 44·16-s + 124·17-s + 6·18-s − 72·19-s − 20·20-s + 40·21-s − 98·23-s − 64·24-s − 55·25-s + 160·26-s − 34·27-s + 200·28-s − 144·29-s + 8·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s − 5/4·4-s + 0.178·5-s + 0.272·6-s − 1.07·7-s + 1.41·8-s − 1/9·9-s − 0.126·10-s + 0.481·12-s − 1.70·13-s + 0.763·14-s − 0.0688·15-s + 0.687·16-s + 1.76·17-s + 0.0785·18-s − 0.869·19-s − 0.223·20-s + 0.415·21-s − 0.888·23-s − 0.544·24-s − 0.439·25-s + 1.20·26-s − 0.242·27-s + 1.34·28-s − 0.922·29-s + 0.0486·30-s + ⋯

Functional equation

Λ(s)=(14641s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(14641s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1464114641    =    11411^{4}
Sign: 11
Analytic conductor: 50.968650.9686
Root analytic conductor: 2.671932.67193
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 14641, ( :3/2,3/2), 1)(4,\ 14641,\ (\ :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)
bad11 1 1
good2D4D_{4} 1+pT+7pT2+p4T3+p6T4 1 + p T + 7 p T^{2} + p^{4} T^{3} + p^{6} T^{4}
3D4D_{4} 1+2T+7T2+2p3T3+p6T4 1 + 2 T + 7 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1−2T+59T2−2p3T3+p6T4 1 - 2 T + 59 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+20T+738T2+20p3T3+p6T4 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+80T+4794T2+80p3T3+p6T4 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−124T+13238T2−124p3T3+p6T4 1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+72T+4214T2+72p3T3+p6T4 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+98T+22847T2+98p3T3+p6T4 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+144T+44554T2+144p3T3+p6T4 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+34T+57519T2+34p3T3+p6T4 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−54T+101843T2−54p3T3+p6T4 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+536T+209618T2+536p3T3+p6T4 1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−60T+159146T2−60p3T3+p6T4 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+272T+182942T2+272p3T3+p6T4 1 + 272 T + 182942 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+492T+348862T2+492p3T3+p6T4 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−634T+458975T2−634p3T3+p6T4 1 - 634 T + 458975 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+840T+528794T2+840p3T3+p6T4 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−754T+742455T2−754p3T3+p6T4 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+678T+813415T2+678p3T3+p6T4 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−400T+160962T2−400p3T3+p6T4 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+4pT−279966T2+4p4T3+p6T4 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}
83D4D_{4} 1+468T+1155130T2+468p3T3+p6T4 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1842T+1935427T2+1842p3T3+p6T4 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−2194T+2966547T2−2194p3T3+p6T4 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4}
show more
show less
   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

−12.60049138921098189658558253409, −12.39583005398207801588432958358, −11.80894701207969181601106078217, −11.06214396111645016321095234316, −10.22117977230772887434677405655, −9.901678808544750095756864408089, −9.661350687673465733032615363812, −9.305897447054092520175674310873, −8.295524804042286583231505106990, −8.095185549472386784858743798541, −7.35543080829368817035264404875, −6.68386104755497933577516822007, −5.82553932576668162270287513298, −5.32280648645674888756322103045, −4.65150366772600906780035440712, −3.83879729854836863296002502972, −3.05088622613104236600227662442, −1.66692501185640243428159567255, 0, 0, 1.66692501185640243428159567255, 3.05088622613104236600227662442, 3.83879729854836863296002502972, 4.65150366772600906780035440712, 5.32280648645674888756322103045, 5.82553932576668162270287513298, 6.68386104755497933577516822007, 7.35543080829368817035264404875, 8.095185549472386784858743798541, 8.295524804042286583231505106990, 9.305897447054092520175674310873, 9.661350687673465733032615363812, 9.901678808544750095756864408089, 10.22117977230772887434677405655, 11.06214396111645016321095234316, 11.80894701207969181601106078217, 12.39583005398207801588432958358, 12.60049138921098189658558253409

Graph of the ZZ-function along the critical line