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Properties

Label 8-6e16-1.1-c2e4-0-31
Degree 88
Conductor 2.821×10122.821\times 10^{12}
Sign 11
Analytic cond. 1.55510×1061.55510\times 10^{6}
Root an. cond. 5.942515.94251
Motivic weight 22
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  + 4·7-s + 20·13-s + 40·19-s + 46·25-s + 76·31-s + 128·37-s − 92·43-s − 78·49-s − 124·61-s − 212·67-s − 208·73-s + 28·79-s + 80·91-s − 28·97-s + 148·103-s − 64·109-s + 394·121-s + 127-s + 131-s + 160·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4/7·7-s + 1.53·13-s + 2.10·19-s + 1.83·25-s + 2.45·31-s + 3.45·37-s − 2.13·43-s − 1.59·49-s − 2.03·61-s − 3.16·67-s − 2.84·73-s + 0.354·79-s + 0.879·91-s − 0.288·97-s + 1.43·103-s − 0.587·109-s + 3.25·121-s + 0.00787·127-s + 0.00763·131-s + 1.20·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

Λ(s)=((216⋅316)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅316)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅3162^{16} \cdot 3^{16}
Sign: 11
Analytic conductor: 1.55510×1061.55510\times 10^{6}
Root analytic conductor: 5.942515.94251
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅316, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{16} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 10.3672025210.36720252
L(12)L(\frac12) ≈\approx 10.3672025210.36720252
L(2)L(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
good5C22C_2^2 (1−23T2+p4T4)2 ( 1 - 23 T^{2} + p^{4} T^{4} )^{2}
7D4D_{4} (1−2T+45T2−2p2T3+p4T4)2 ( 1 - 2 T + 45 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
11D4×C2D_4\times C_2 1−394T2+66147T4−394p4T6+p8T8 1 - 394 T^{2} + 66147 T^{4} - 394 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1−10T+147T2−10p2T3+p4T4)2 ( 1 - 10 T + 147 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4×C2D_4\times C_2 1−796T2+294342T4−796p4T6+p8T8 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1−20T+606T2−20p2T3+p4T4)2 ( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1−2026T2+1583907T4−2026p4T6+p8T8 1 - 2026 T^{2} + 1583907 T^{4} - 2026 p^{4} T^{6} + p^{8} T^{8}
29D4×C2D_4\times C_2 1−3166T2+3912675T4−3166p4T6+p8T8 1 - 3166 T^{2} + 3912675 T^{4} - 3166 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1−38T+1797T2−38p2T3+p4T4)2 ( 1 - 38 T + 1797 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2}
37D4D_{4} (1−64T+3546T2−64p2T3+p4T4)2 ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4×C2D_4\times C_2 1−2782T2+4157187T4−2782p4T6+p8T8 1 - 2782 T^{2} + 4157187 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1+46T+87pT2+46p2T3+p4T4)2 ( 1 + 46 T + 87 p T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−6586T2+19745907T4−6586p4T6+p8T8 1 - 6586 T^{2} + 19745907 T^{4} - 6586 p^{4} T^{6} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−2236T2−2409114T4−2236p4T6+p8T8 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−5194T2+14880867T4−5194p4T6+p8T8 1 - 5194 T^{2} + 14880867 T^{4} - 5194 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+62T+6459T2+62p2T3+p4T4)2 ( 1 + 62 T + 6459 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4D_{4} (1+106T+11301T2+106p2T3+p4T4)2 ( 1 + 106 T + 11301 T^{2} + 106 p^{2} T^{3} + p^{4} T^{4} )^{2}
71D4×C2D_4\times C_2 1−12460T2+77194662T4−12460p4T6+p8T8 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+104T+11418T2+104p2T3+p4T4)2 ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4D_{4} (1−14T+11181T2−14p2T3+p4T4)2 ( 1 - 14 T + 11181 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4×C2D_4\times C_2 1−2842T2+75503283T4−2842p4T6+p8T8 1 - 2842 T^{2} + 75503283 T^{4} - 2842 p^{4} T^{6} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−8860T2+51019782T4−8860p4T6+p8T8 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8}
97D4D_{4} (1+14T+8283T2+14p2T3+p4T4)2 ( 1 + 14 T + 8283 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
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   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

−6.63674795616273895876505659725, −6.42479732401661562307382833439, −6.19606488363671555771599339384, −6.02930158113953161419912389683, −5.96940706798490210695117946167, −5.60357807915222880898435233247, −5.45078866833320293781740234946, −4.94997701080550235673402033966, −4.92721674740632177856721791498, −4.60477279413183678912692375328, −4.42843610949763959023458787702, −4.35137207489972473608390977251, −4.15019956964612132070685409468, −3.47557114106265314696308564854, −3.32062594632506754131328687417, −3.12099422938010203503235940134, −2.94982430615021434405385558993, −2.82858162081145648568382306321, −2.48722870255257886592539478473, −1.71761702231169970825154405321, −1.70575107382543915709252610797, −1.33684776117530933015748656369, −1.12732748443671279187222178068, −0.64340256405058581337037716440, −0.51741568217380455299230901792, 0.51741568217380455299230901792, 0.64340256405058581337037716440, 1.12732748443671279187222178068, 1.33684776117530933015748656369, 1.70575107382543915709252610797, 1.71761702231169970825154405321, 2.48722870255257886592539478473, 2.82858162081145648568382306321, 2.94982430615021434405385558993, 3.12099422938010203503235940134, 3.32062594632506754131328687417, 3.47557114106265314696308564854, 4.15019956964612132070685409468, 4.35137207489972473608390977251, 4.42843610949763959023458787702, 4.60477279413183678912692375328, 4.92721674740632177856721791498, 4.94997701080550235673402033966, 5.45078866833320293781740234946, 5.60357807915222880898435233247, 5.96940706798490210695117946167, 6.02930158113953161419912389683, 6.19606488363671555771599339384, 6.42479732401661562307382833439, 6.63674795616273895876505659725

Graph of the ZZ-function along the critical line