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Properties

Label 8-2160e4-1.1-c0e4-0-1
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.350341.35034
Root an. cond. 1.038251.03825
Motivic weight 00
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  − 4·19-s + 2·49-s − 4·61-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·19-s + 2·49-s − 4·61-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)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)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s)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)^{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: 1.350341.35034
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :0,0,0,0), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.81504956500.8150495650
L(12)L(\frac12) ≈\approx 0.81504956500.8150495650
L(1)L(1) 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
5C22C_2^2 1+T4 1 + T^{4}
good7C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
11C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
13C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
17C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
19C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
23C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
29C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
31C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
37C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
41C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
43C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
47C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
53C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
59C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
61C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
67C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
71C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
73C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
79C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + 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

−6.51238460270033905146623495305, −6.50694258561531982445029001036, −6.50347751916732354917210509194, −5.99004745699615919943991739275, −5.89068762861157760291461271730, −5.63076908321124070471609486504, −5.38412833681878083567992811931, −5.33888141925683638682176174584, −4.71388620877052884621676361421, −4.67362916347042190253038986671, −4.47748150253517346951882482002, −4.46596650223500158213644188886, −4.12912090296278936926376274417, −3.81147327691642548807846911105, −3.68493575279538663600998276755, −3.39341794277616475597240913041, −3.13525931027800871266299781335, −2.79236654733543617383755313581, −2.43612157846312220841865952203, −2.40825720303266386130029002400, −1.95934878714665840016153863370, −1.86200313568443260690063170890, −1.58062512804029687945387432507, −0.994737123430730438835313106031, −0.46525003722821184951591825544, 0.46525003722821184951591825544, 0.994737123430730438835313106031, 1.58062512804029687945387432507, 1.86200313568443260690063170890, 1.95934878714665840016153863370, 2.40825720303266386130029002400, 2.43612157846312220841865952203, 2.79236654733543617383755313581, 3.13525931027800871266299781335, 3.39341794277616475597240913041, 3.68493575279538663600998276755, 3.81147327691642548807846911105, 4.12912090296278936926376274417, 4.46596650223500158213644188886, 4.47748150253517346951882482002, 4.67362916347042190253038986671, 4.71388620877052884621676361421, 5.33888141925683638682176174584, 5.38412833681878083567992811931, 5.63076908321124070471609486504, 5.89068762861157760291461271730, 5.99004745699615919943991739275, 6.50347751916732354917210509194, 6.50694258561531982445029001036, 6.51238460270033905146623495305

Graph of the ZZ-function along the critical line