↑

Properties

Label 8-2160e4-1.1-c2e4-0-12
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.19992×1071.19992\times 10^{7}
Root an. cond. 7.671747.67174
Motivic weight 22
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  + 88·13-s + 10·25-s + 8·37-s + 76·49-s − 124·61-s + 304·73-s − 128·97-s − 812·109-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.16e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 6.76·13-s + 2/5·25-s + 8/37·37-s + 1.55·49-s − 2.03·61-s + 4.16·73-s − 1.31·97-s − 7.44·109-s + 3.20·121-s + 0.00787·127-s + 0.00763·131-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 + 24.6·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−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(3-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1)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+1)^{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: 1.19992×1071.19992\times 10^{7}
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 16.0223390816.02233908
L(12)L(\frac12) ≈\approx 16.0223390816.02233908
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
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
good7C22C_2^2 (1−38T2+p4T4)2 ( 1 - 38 T^{2} + p^{4} T^{4} )^{2}
11C22C_2^2 (1−194T2+p4T4)2 ( 1 - 194 T^{2} + p^{4} T^{4} )^{2}
13C2C_2 (1−22T+p2T2)4 ( 1 - 22 T + p^{2} T^{2} )^{4}
17C22C_2^2 (1+533T2+p4T4)2 ( 1 + 533 T^{2} + p^{4} T^{4} )^{2}
19C22C_2^2 (1+13T2+p4T4)2 ( 1 + 13 T^{2} + p^{4} T^{4} )^{2}
23C22C_2^2 (1−191T2+p4T4)2 ( 1 - 191 T^{2} + p^{4} T^{4} )^{2}
29C22C_2^2 (1+62T2+p4T4)2 ( 1 + 62 T^{2} + p^{4} T^{4} )^{2}
31C22C_2^2 (1−1547T2+p4T4)2 ( 1 - 1547 T^{2} + p^{4} T^{4} )^{2}
37C2C_2 (1−2T+p2T2)4 ( 1 - 2 T + p^{2} T^{2} )^{4}
41C22C_2^2 (1+482T2+p4T4)2 ( 1 + 482 T^{2} + p^{4} T^{4} )^{2}
43C22C_2^2 (1−3458T2+p4T4)2 ( 1 - 3458 T^{2} + p^{4} T^{4} )^{2}
47C22C_2^2 (1−4226T2+p4T4)2 ( 1 - 4226 T^{2} + p^{4} T^{4} )^{2}
53C22C_2^2 (1+5573T2+p4T4)2 ( 1 + 5573 T^{2} + p^{4} T^{4} )^{2}
59C22C_2^2 (1−6374T2+p4T4)2 ( 1 - 6374 T^{2} + p^{4} T^{4} )^{2}
61C2C_2 (1+31T+p2T2)4 ( 1 + 31 T + p^{2} T^{2} )^{4}
67C22C_2^2 (1−8438T2+p4T4)2 ( 1 - 8438 T^{2} + p^{4} T^{4} )^{2}
71C22C_2^2 (1+2206T2+p4T4)2 ( 1 + 2206 T^{2} + p^{4} T^{4} )^{2}
73C2C_2 (1−76T+p2T2)4 ( 1 - 76 T + p^{2} T^{2} )^{4}
79C22C_2^2 (1−12107T2+p4T4)2 ( 1 - 12107 T^{2} + p^{4} T^{4} )^{2}
83C22C_2^2 (1+3097T2+p4T4)2 ( 1 + 3097 T^{2} + p^{4} T^{4} )^{2}
89C22C_2^2 (1+12962T2+p4T4)2 ( 1 + 12962 T^{2} + p^{4} T^{4} )^{2}
97C2C_2 (1+32T+p2T2)4 ( 1 + 32 T + p^{2} 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

−6.48531317644886442151920960884, −6.07542827416156136072810739620, −5.72279026492969566119567566357, −5.59331053482706433376081585094, −5.55999800126947305266471731331, −5.51078574170880522440764660094, −5.13039237013168219602335900508, −4.56132097123238993334321946244, −4.50289393042389497749177703367, −4.15945081184647979171306654079, −4.03423783065252859800427068087, −3.92465945451352093478958912425, −3.65587893003371432876768009402, −3.34257168041660878304482728432, −3.32690525519222437504693488878, −3.11679044047596477084062137919, −2.76823742969333761932864280262, −2.25917618536470218945845711803, −2.16377751141339772657054805243, −1.59680169695039543470418274483, −1.39575439400212836757417761253, −1.25561480993411243329743980707, −1.12879341483121352417393478149, −0.56712017170961588831119869205, −0.54245034668820873482215305507, 0.54245034668820873482215305507, 0.56712017170961588831119869205, 1.12879341483121352417393478149, 1.25561480993411243329743980707, 1.39575439400212836757417761253, 1.59680169695039543470418274483, 2.16377751141339772657054805243, 2.25917618536470218945845711803, 2.76823742969333761932864280262, 3.11679044047596477084062137919, 3.32690525519222437504693488878, 3.34257168041660878304482728432, 3.65587893003371432876768009402, 3.92465945451352093478958912425, 4.03423783065252859800427068087, 4.15945081184647979171306654079, 4.50289393042389497749177703367, 4.56132097123238993334321946244, 5.13039237013168219602335900508, 5.51078574170880522440764660094, 5.55999800126947305266471731331, 5.59331053482706433376081585094, 5.72279026492969566119567566357, 6.07542827416156136072810739620, 6.48531317644886442151920960884

Graph of the ZZ-function along the critical line