Properties

Label 8-1840e4-1.1-c1e4-0-3
Degree 88
Conductor 1.146×10131.146\times 10^{13}
Sign 11
Analytic cond. 46599.346599.3
Root an. cond. 3.833073.83307
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  + 8·9-s − 10·25-s − 24·29-s + 48·41-s − 8·49-s + 30·81-s + 72·101-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 8/3·9-s − 2·25-s − 4.45·29-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 7.16·101-s − 4·121-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 + 4·169-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 + 0.0669·223-s + ⋯

Functional equation

Λ(s)=((21654234)s/2ΓC(s)4L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((21654234)s/2ΓC(s+1/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216542342^{16} \cdot 5^{4} \cdot 23^{4}
Sign: 11
Analytic conductor: 46599.346599.3
Root analytic conductor: 3.833073.83307
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 21654234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) \approx 3.1690885003.169088500
L(12)L(\frac12) \approx 3.1690885003.169088500
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 144T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
good3C22C_2^2 (14T2+p2T4)2 ( 1 - 4 T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2 (1+4T2+p2T4)2 ( 1 + 4 T^{2} + p^{2} T^{4} )^{2}
11C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
13C2C_2 (1pT2)4 ( 1 - p T^{2} )^{4}
17C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
19C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
29C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
31C2C_2 (1pT2)4 ( 1 - p T^{2} )^{4}
37C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
41C2C_2 (112T+pT2)4 ( 1 - 12 T + p T^{2} )^{4}
43C22C_2^2 (1+76T2+p2T4)2 ( 1 + 76 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+4T2+p2T4)2 ( 1 + 4 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
59C2C_2 (1pT2)4 ( 1 - p T^{2} )^{4}
61C2C_2 (18T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
67C22C_2^2 (1116T2+p2T4)2 ( 1 - 116 T^{2} + p^{2} T^{4} )^{2}
71C2C_2 (1pT2)4 ( 1 - p T^{2} )^{4}
73C2C_2 (1pT2)4 ( 1 - p T^{2} )^{4}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C22C_2^2 (1+76T2+p2T4)2 ( 1 + 76 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (16T+pT2)2(1+6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
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   L(s)=p j=18(1αj,pps)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

−6.53030672977214159267916621211, −6.33830228616134470610451434397, −6.05332309495927469186098916214, −5.95054161269808339431890328114, −5.87828133552865796052554823830, −5.49144701961845764894916238254, −5.44125879953846344446454867939, −4.98576917281967103727691240206, −4.94654517481940278330848550806, −4.32511251589903727092955923506, −4.24622284453921309041975398225, −4.22353258160984330676185182317, −4.07293377587954730748701419088, −3.97095303718592121737034638264, −3.36165500083336533606413305160, −3.29590481158386921081225895181, −3.20672562077088951890911286649, −2.35789569964417640388409082685, −2.26038205265691797041900241180, −2.12864841305706741144718164091, −1.98267923644825071147193256303, −1.50523716604757432411317030971, −1.05077269191663283650414639431, −0.992719158316320777705041565287, −0.31706902330945866442491259851, 0.31706902330945866442491259851, 0.992719158316320777705041565287, 1.05077269191663283650414639431, 1.50523716604757432411317030971, 1.98267923644825071147193256303, 2.12864841305706741144718164091, 2.26038205265691797041900241180, 2.35789569964417640388409082685, 3.20672562077088951890911286649, 3.29590481158386921081225895181, 3.36165500083336533606413305160, 3.97095303718592121737034638264, 4.07293377587954730748701419088, 4.22353258160984330676185182317, 4.24622284453921309041975398225, 4.32511251589903727092955923506, 4.94654517481940278330848550806, 4.98576917281967103727691240206, 5.44125879953846344446454867939, 5.49144701961845764894916238254, 5.87828133552865796052554823830, 5.95054161269808339431890328114, 6.05332309495927469186098916214, 6.33830228616134470610451434397, 6.53030672977214159267916621211

Graph of the ZZ-function along the critical line