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Properties

Label 8-2160e4-1.1-c1e4-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·13-s − 2·25-s + 8·37-s + 28·49-s + 8·61-s − 32·73-s − 8·97-s + 40·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 198·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 5.54·13-s − 2/5·25-s + 1.31·37-s + 4·49-s + 1.02·61-s − 3.74·73-s − 0.812·97-s + 3.83·109-s − 3.45·121-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 + 15.2·169-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 0.79268544020.7926854402
L(12)L(\frac12) ≈\approx 0.79268544020.7926854402
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+19T2+p2T4)2 ( 1 + 19 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1+5T+pT2)4 ( 1 + 5 T + p T^{2} )^{4}
17C22C_2^2 (1−25T2+p2T4)2 ( 1 - 25 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
23C22C_2^2 (1−29T2+p2T4)2 ( 1 - 29 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+23T2+p2T4)2 ( 1 + 23 T^{2} + p^{2} T^{4} )^{2}
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 (1−11T2+p2T4)2 ( 1 - 11 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+91T2+p2T4)2 ( 1 + 91 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+106T2+p2T4)2 ( 1 + 106 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−2T+pT2)4 ( 1 - 2 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 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1+8T+pT2)4 ( 1 + 8 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+58T2+p2T4)2 ( 1 + 58 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
97C2C_2 (1+2T+pT2)4 ( 1 + 2 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

−6.42639464090257224362096110031, −6.15956547338578833517903868716, −5.99004628223642233969274527716, −5.77878425001991175686669282502, −5.62154293604148583925430278153, −5.34438104874535338444917183643, −5.05162141564394852685332829476, −5.04776956249746308902006493515, −4.67876519444255430857859756633, −4.60842711787946228282542843402, −4.48745386809439008593739327482, −4.06615573386152862034482968478, −3.97072032719299627320231334092, −3.69586874581023242496531991373, −3.31348321307895913107532656244, −2.87262717924069307700139240477, −2.78799980749226680917202734656, −2.46744624116600037250749210666, −2.42833168088655551924914365426, −2.27770644008203055420211636656, −1.99314528461951552827039882426, −1.42095124370104356387348182277, −1.11998333996967552298160488249, −0.49965363800857599949030033960, −0.23238267927961133255810148763, 0.23238267927961133255810148763, 0.49965363800857599949030033960, 1.11998333996967552298160488249, 1.42095124370104356387348182277, 1.99314528461951552827039882426, 2.27770644008203055420211636656, 2.42833168088655551924914365426, 2.46744624116600037250749210666, 2.78799980749226680917202734656, 2.87262717924069307700139240477, 3.31348321307895913107532656244, 3.69586874581023242496531991373, 3.97072032719299627320231334092, 4.06615573386152862034482968478, 4.48745386809439008593739327482, 4.60842711787946228282542843402, 4.67876519444255430857859756633, 5.04776956249746308902006493515, 5.05162141564394852685332829476, 5.34438104874535338444917183643, 5.62154293604148583925430278153, 5.77878425001991175686669282502, 5.99004628223642233969274527716, 6.15956547338578833517903868716, 6.42639464090257224362096110031

Graph of the ZZ-function along the critical line