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Properties

Label 8-2160e4-1.1-c1e4-0-20
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·13-s − 2·25-s + 8·37-s + 28·49-s + 44·61-s + 40·73-s + 64·97-s + 4·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4.43·13-s − 2/5·25-s + 1.31·37-s + 4·49-s + 5.63·61-s + 4.68·73-s + 6.49·97-s + 0.383·109-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−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(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)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+1/2)^{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: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 10.2982684510.29826845
L(12)L(\frac12) ≈\approx 10.2982684510.29826845
L(32)L(\frac{3}{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+T2)2 ( 1 + T^{2} )^{2}
good7C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
11C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
17C22C_2^2 (1−25T2+p2T4)2 ( 1 - 25 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−7T+pT2)2(1+7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}
23C22C_2^2 (1+43T2+p2T4)2 ( 1 + 43 T^{2} + p^{2} T^{4} )^{2}
29C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
31C2C_2 (1−11T+pT2)2(1+11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
41C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−74T2+p2T4)2 ( 1 - 74 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+82T2+p2T4)2 ( 1 + 82 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−97T2+p2T4)2 ( 1 - 97 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+70T2+p2T4)2 ( 1 + 70 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−11T+pT2)4 ( 1 - 11 T + p T^{2} )^{4}
67C2C_2 (1−16T+pT2)2(1+16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}
71C22C_2^2 (1+130T2+p2T4)2 ( 1 + 130 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
79C2C_2 (1−13T+pT2)2(1+13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}
83C22C_2^2 (1+139T2+p2T4)2 ( 1 + 139 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+146T2+p2T4)2 ( 1 + 146 T^{2} + p^{2} T^{4} )^{2}
97C2C_2 (1−16T+pT2)4 ( 1 - 16 T + p T^{2} )^{4}
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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

−6.46475564136412098674118331359, −6.11472519359885869200352135259, −6.06250451855410475564519498036, −5.84956113770167196143365289265, −5.81181949498335120766894350145, −5.30512928718617604262627456682, −5.26143362617949643855069495526, −5.09362433305188909596161715492, −4.91019850318182765546988005680, −4.19466329616951735462025244542, −4.11888334293400507511717988645, −4.08616806344695161465742946602, −3.98556232353403183463140996509, −3.55019434004427301762737108254, −3.44357412037090026222253216604, −3.31611044809946375424304163356, −3.05289261089256111662310783569, −2.29055264224896745237293799100, −2.25045616571239268310439312535, −2.21652973268923967098211629542, −1.94329112336789245849960624355, −1.08967268402571049946421798937, −1.05433488004310442440802175617, −0.879090882508719592218855567074, −0.68750312518704842498304116880, 0.68750312518704842498304116880, 0.879090882508719592218855567074, 1.05433488004310442440802175617, 1.08967268402571049946421798937, 1.94329112336789245849960624355, 2.21652973268923967098211629542, 2.25045616571239268310439312535, 2.29055264224896745237293799100, 3.05289261089256111662310783569, 3.31611044809946375424304163356, 3.44357412037090026222253216604, 3.55019434004427301762737108254, 3.98556232353403183463140996509, 4.08616806344695161465742946602, 4.11888334293400507511717988645, 4.19466329616951735462025244542, 4.91019850318182765546988005680, 5.09362433305188909596161715492, 5.26143362617949643855069495526, 5.30512928718617604262627456682, 5.81181949498335120766894350145, 5.84956113770167196143365289265, 6.06250451855410475564519498036, 6.11472519359885869200352135259, 6.46475564136412098674118331359

Graph of the ZZ-function along the critical line