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Properties

Label 8-960e4-1.1-c1e4-0-18
Degree 88
Conductor 849346560000849346560000
Sign 11
Analytic cond. 3452.973452.97
Root an. cond. 2.768682.76868
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  + 6·9-s − 16·13-s − 2·25-s + 16·37-s + 4·49-s + 40·61-s − 8·73-s + 27·81-s + 40·97-s − 40·109-s − 96·117-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 + ⋯
L(s)  = 1  + 2·9-s − 4.43·13-s − 2/5·25-s + 2.63·37-s + 4/7·49-s + 5.12·61-s − 0.936·73-s + 3·81-s + 4.06·97-s − 3.83·109-s − 8.87·117-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 + ⋯

Functional equation

Λ(s)=((224⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \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: 224⋅34⋅542^{24} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 3452.973452.97
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.6760764292.676076429
L(12)L(\frac12) ≈\approx 2.6760764292.676076429
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
3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
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+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−8T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
23C22C_2^2 (1+34T2+p2T4)2 ( 1 + 34 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
41C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 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 (1−70T2+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 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
67C22C_2^2 (1−86T2+p2T4)2 ( 1 - 86 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
79C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{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 (1−10T+pT2)4 ( 1 - 10 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

−7.11497971684968336468316248350, −6.99383880967085274439688156584, −6.82580653174917820999052156720, −6.69837545135997129217770093922, −6.50209597120957223609352888581, −5.91290086632325466704977949110, −5.81992161310353618861476197595, −5.59029648855747847446993202119, −5.11488404002943330331590815223, −5.01015137904076881958405818697, −5.00500413134931241950736903349, −4.62727704328125151185581494693, −4.30505036211368661504363219478, −4.27254604993901083543535084823, −3.89865217245323185442279801352, −3.75193665852114912527313561294, −3.29956783434892709884503642743, −2.74519630418242092018695059058, −2.66395661926230098667515205010, −2.41587507820056681957513959122, −2.06121599581559811849482216811, −2.00009676375686446543975626198, −1.35594937719992004067689183599, −0.76714289168996611660210428909, −0.48937519566131699730116159891, 0.48937519566131699730116159891, 0.76714289168996611660210428909, 1.35594937719992004067689183599, 2.00009676375686446543975626198, 2.06121599581559811849482216811, 2.41587507820056681957513959122, 2.66395661926230098667515205010, 2.74519630418242092018695059058, 3.29956783434892709884503642743, 3.75193665852114912527313561294, 3.89865217245323185442279801352, 4.27254604993901083543535084823, 4.30505036211368661504363219478, 4.62727704328125151185581494693, 5.00500413134931241950736903349, 5.01015137904076881958405818697, 5.11488404002943330331590815223, 5.59029648855747847446993202119, 5.81992161310353618861476197595, 5.91290086632325466704977949110, 6.50209597120957223609352888581, 6.69837545135997129217770093922, 6.82580653174917820999052156720, 6.99383880967085274439688156584, 7.11497971684968336468316248350

Graph of the ZZ-function along the critical line