↑

Properties

Label 8-24e8-1.1-c4e4-0-2
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 1.25680×1071.25680\times 10^{7}
Root an. cond. 7.716287.71628
Motivic weight 44
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  + 1.22e3·17-s + 1.32e3·25-s − 1.18e4·41-s + 2.87e3·49-s − 2.35e4·73-s − 3.50e4·89-s + 2.36e4·97-s − 6.14e4·113-s − 5.81e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.13e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4.23·17-s + 2.11·25-s − 7.06·41-s + 1.19·49-s − 4.42·73-s − 4.42·89-s + 2.51·97-s − 4.80·113-s − 3.97·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.96·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯

Functional equation

Λ(s)=((224⋅38)s/2 ΓC(s)4 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
Λ(s)=((224⋅38)s/2 ΓC(s+2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 1.25680×1071.25680\times 10^{7}
Root analytic conductor: 7.716287.71628
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :2,2,2,2), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.5530190861.553019086
L(12)L(\frac12) ≈\approx 1.5530190861.553019086
L(3)L(3) 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 (1−662T2+p8T4)2 ( 1 - 662 T^{2} + p^{8} T^{4} )^{2}
7C22C_2^2 (1−1438T2+p8T4)2 ( 1 - 1438 T^{2} + p^{8} T^{4} )^{2}
11C22C_2^2 (1+29090T2+p8T4)2 ( 1 + 29090 T^{2} + p^{8} T^{4} )^{2}
13C22C_2^2 (1−56690T2+p8T4)2 ( 1 - 56690 T^{2} + p^{8} T^{4} )^{2}
17C2C_2 (1−18pT+p4T2)4 ( 1 - 18 p T + p^{4} T^{2} )^{4}
19C22C_2^2 (1−102670T2+p8T4)2 ( 1 - 102670 T^{2} + p^{8} T^{4} )^{2}
23C22C_2^2 (1−340658T2+p8T4)2 ( 1 - 340658 T^{2} + p^{8} T^{4} )^{2}
29C22C_2^2 (1+732586T2+p8T4)2 ( 1 + 732586 T^{2} + p^{8} T^{4} )^{2}
31C22C_2^2 (1−1834942T2+p8T4)2 ( 1 - 1834942 T^{2} + p^{8} T^{4} )^{2}
37C22C_2^2 (1−2668322T2+p8T4)2 ( 1 - 2668322 T^{2} + p^{8} T^{4} )^{2}
41C2C_2 (1+2970T+p4T2)4 ( 1 + 2970 T + p^{4} T^{2} )^{4}
43C22C_2^2 (1−1509070T2+p8T4)2 ( 1 - 1509070 T^{2} + p^{8} T^{4} )^{2}
47C22C_2^2 (1−9602546T2+p8T4)2 ( 1 - 9602546 T^{2} + p^{8} T^{4} )^{2}
53C22C_2^2 (1−14513462T2+p8T4)2 ( 1 - 14513462 T^{2} + p^{8} T^{4} )^{2}
59C22C_2^2 (1+17045810T2+p8T4)2 ( 1 + 17045810 T^{2} + p^{8} T^{4} )^{2}
61C22C_2^2 (1+8140126T2+p8T4)2 ( 1 + 8140126 T^{2} + p^{8} T^{4} )^{2}
67C22C_2^2 (1+17250290T2+p8T4)2 ( 1 + 17250290 T^{2} + p^{8} T^{4} )^{2}
71C22C_2^2 (1−7421618T2+p8T4)2 ( 1 - 7421618 T^{2} + p^{8} T^{4} )^{2}
73C2C_2 (1+5894T+p4T2)4 ( 1 + 5894 T + p^{4} T^{2} )^{4}
79C22C_2^2 (1−5887966T2+p8T4)2 ( 1 - 5887966 T^{2} + p^{8} T^{4} )^{2}
83C22C_2^2 (1+94916450T2+p8T4)2 ( 1 + 94916450 T^{2} + p^{8} T^{4} )^{2}
89C2C_2 (1+8766T+p4T2)4 ( 1 + 8766 T + p^{4} T^{2} )^{4}
97C2C_2 (1−5918T+p4T2)4 ( 1 - 5918 T + p^{4} 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

−6.96853020118629211389195352374, −6.89926591036386455210925821746, −6.74935382347212281887900576114, −6.52685681212796169166746256605, −6.05334778555968845348108606937, −5.75912214044975777687463040411, −5.71707568254131179888104616009, −5.19108827014840352275216159500, −5.06387795348541500967621120396, −5.05039813869849369966718501602, −5.02888363041396077347805217448, −4.19854130125694695183558640409, −3.93296366845340768884598355175, −3.87807082957901220294997195080, −3.43941429242788714875623886050, −3.07772995700007501562644237282, −3.05382735755918681139550269290, −2.81874914104307685120001305152, −2.50603213485441464834279846211, −1.64889686292710059698867690061, −1.43059986297073727279643692032, −1.34453268586427533506611922395, −1.30226530012618004123728893309, −0.53589296286443654262950760966, −0.15988433332669184288079933666, 0.15988433332669184288079933666, 0.53589296286443654262950760966, 1.30226530012618004123728893309, 1.34453268586427533506611922395, 1.43059986297073727279643692032, 1.64889686292710059698867690061, 2.50603213485441464834279846211, 2.81874914104307685120001305152, 3.05382735755918681139550269290, 3.07772995700007501562644237282, 3.43941429242788714875623886050, 3.87807082957901220294997195080, 3.93296366845340768884598355175, 4.19854130125694695183558640409, 5.02888363041396077347805217448, 5.05039813869849369966718501602, 5.06387795348541500967621120396, 5.19108827014840352275216159500, 5.71707568254131179888104616009, 5.75912214044975777687463040411, 6.05334778555968845348108606937, 6.52685681212796169166746256605, 6.74935382347212281887900576114, 6.89926591036386455210925821746, 6.96853020118629211389195352374

Graph of the ZZ-function along the critical line