↑

Properties

Label 8-2160e4-1.1-c3e4-0-1
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 2.63802×1082.63802\times 10^{8}
Root an. cond. 11.289111.2891
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 136·13-s − 50·25-s − 136·37-s − 572·49-s + 1.96e3·61-s − 1.32e3·73-s + 880·97-s + 7.63e3·109-s − 620·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.77e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2.90·13-s − 2/5·25-s − 0.604·37-s − 1.66·49-s + 4.12·61-s − 2.12·73-s + 0.921·97-s + 6.71·109-s − 0.465·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.26·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 2.63802×1082.63802\times 10^{8}
Root analytic conductor: 11.289111.2891
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 10.3207089510.32070895
L(12)L(\frac12) ≈\approx 10.3207089510.32070895
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
3 1 1
5C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
good7C2C_2 (1−20T+p3T2)2(1+20T+p3T2)2 ( 1 - 20 T + p^{3} T^{2} )^{2}( 1 + 20 T + p^{3} T^{2} )^{2}
11C22C_2^2 (1+310T2+p6T4)2 ( 1 + 310 T^{2} + p^{6} T^{4} )^{2}
13C2C_2 (1−34T+p3T2)4 ( 1 - 34 T + p^{3} T^{2} )^{4}
17C22C_2^2 (1−9817T2+p6T4)2 ( 1 - 9817 T^{2} + p^{6} T^{4} )^{2}
19C22C_2^2 (1+565T2+p6T4)2 ( 1 + 565 T^{2} + p^{6} T^{4} )^{2}
23C22C_2^2 (1+15259T2+p6T4)2 ( 1 + 15259 T^{2} + p^{6} T^{4} )^{2}
29C22C_2^2 (1−32902T2+p6T4)2 ( 1 - 32902 T^{2} + p^{6} T^{4} )^{2}
31C22C_2^2 (1−27755T2+p6T4)2 ( 1 - 27755 T^{2} + p^{6} T^{4} )^{2}
37C2C_2 (1+34T+p3T2)4 ( 1 + 34 T + p^{3} T^{2} )^{4}
41C22C_2^2 (1−47842T2+p6T4)2 ( 1 - 47842 T^{2} + p^{6} T^{4} )^{2}
43C22C_2^2 (1+56458T2+p6T4)2 ( 1 + 56458 T^{2} + p^{6} T^{4} )^{2}
47C22C_2^2 (1+1714T2+p6T4)2 ( 1 + 1714 T^{2} + p^{6} T^{4} )^{2}
53C22C_2^2 (1−266425T2+p6T4)2 ( 1 - 266425 T^{2} + p^{6} T^{4} )^{2}
59C22C_2^2 (1+185530T2+p6T4)2 ( 1 + 185530 T^{2} + p^{6} T^{4} )^{2}
61C2C_2 (1−491T+p3T2)4 ( 1 - 491 T + p^{3} T^{2} )^{4}
67C2C_2 (1−16pT+p3T2)2(1+16pT+p3T2)2 ( 1 - 16 p T + p^{3} T^{2} )^{2}( 1 + 16 p T + p^{3} T^{2} )^{2}
71C22C_2^2 (1+318334T2+p6T4)2 ( 1 + 318334 T^{2} + p^{6} T^{4} )^{2}
73C2C_2 (1+332T+p3T2)4 ( 1 + 332 T + p^{3} T^{2} )^{4}
79C22C_2^2 (1−568691T2+p6T4)2 ( 1 - 568691 T^{2} + p^{6} T^{4} )^{2}
83C22C_2^2 (1+426211T2+p6T4)2 ( 1 + 426211 T^{2} + p^{6} T^{4} )^{2}
89C22C_2^2 (1−1078162T2+p6T4)2 ( 1 - 1078162 T^{2} + p^{6} T^{4} )^{2}
97C2C_2 (1−220T+p3T2)4 ( 1 - 220 T + p^{3} T^{2} )^{4}
show more
show less
   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.09613940057447487734595168147, −5.74787454892961606393451294179, −5.65618280374295592788796641461, −5.59925207376098885779685272937, −5.49788852837141822992749053261, −4.82631638637749438830547276805, −4.73949415897019639968202398453, −4.61630544035872720539905142596, −4.55621522498966696102235900688, −3.98778587672961202349029275840, −3.75438954430623237107791584446, −3.72619475149358971151461828209, −3.55737749527986171978653466804, −3.31127117298581055638806351747, −3.08809456885638032217718731853, −2.68297158684374484961690943871, −2.57326942431656596688161863464, −2.10776770179782581120239491950, −1.77104423473474554280871243872, −1.68669400458992056809699039025, −1.56980052703751923465001275090, −0.981143262707105799552505719427, −0.78287514180235501315856590759, −0.59887525882974462965664434984, −0.32974355411708018458060083078, 0.32974355411708018458060083078, 0.59887525882974462965664434984, 0.78287514180235501315856590759, 0.981143262707105799552505719427, 1.56980052703751923465001275090, 1.68669400458992056809699039025, 1.77104423473474554280871243872, 2.10776770179782581120239491950, 2.57326942431656596688161863464, 2.68297158684374484961690943871, 3.08809456885638032217718731853, 3.31127117298581055638806351747, 3.55737749527986171978653466804, 3.72619475149358971151461828209, 3.75438954430623237107791584446, 3.98778587672961202349029275840, 4.55621522498966696102235900688, 4.61630544035872720539905142596, 4.73949415897019639968202398453, 4.82631638637749438830547276805, 5.49788852837141822992749053261, 5.59925207376098885779685272937, 5.65618280374295592788796641461, 5.74787454892961606393451294179, 6.09613940057447487734595168147

Graph of the ZZ-function along the critical line