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Properties

Label 8-975e4-1.1-c1e4-0-17
Degree 88
Conductor 903687890625903687890625
Sign 11
Analytic cond. 3673.893673.89
Root an. cond. 2.790232.79023
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  + 4·3-s − 4·7-s + 6·9-s + 8·13-s − 16-s + 4·19-s − 16·21-s − 4·27-s − 20·31-s − 4·37-s + 32·39-s − 4·48-s + 8·49-s + 16·57-s + 32·61-s − 24·63-s + 20·67-s − 4·73-s − 40·79-s − 37·81-s − 32·91-s − 80·93-s − 28·97-s + 4·109-s − 16·111-s + 4·112-s + 48·117-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.51·7-s + 2·9-s + 2.21·13-s − 1/4·16-s + 0.917·19-s − 3.49·21-s − 0.769·27-s − 3.59·31-s − 0.657·37-s + 5.12·39-s − 0.577·48-s + 8/7·49-s + 2.11·57-s + 4.09·61-s − 3.02·63-s + 2.44·67-s − 0.468·73-s − 4.50·79-s − 4.11·81-s − 3.35·91-s − 8.29·93-s − 2.84·97-s + 0.383·109-s − 1.51·111-s + 0.377·112-s + 4.43·117-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅1343^{4} \cdot 5^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 3673.893673.89
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.6770044402.677004440
L(12)L(\frac12) ≈\approx 2.6770044402.677004440
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)
bad3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
5 1 1
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
good2C23C_2^3 1+T4+p4T8 1 + T^{4} + p^{4} T^{8}
7C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
11C23C_2^3 1−206T4+p4T8 1 - 206 T^{4} + p^{4} T^{8}
17C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
19C22C_2^2 (1−2T+2T2−2pT3+p2T4)2 ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+10T+50T2+10pT3+p2T4)2 ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
37C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
41C23C_2^3 1+2722T4+p4T8 1 + 2722 T^{4} + p^{4} T^{8}
43C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+1666T4+p4T8 1 + 1666 T^{4} + p^{4} T^{8}
53C22C_2^2 (1−74T2+p2T4)2 ( 1 - 74 T^{2} + p^{2} T^{4} )^{2}
59C23C_2^3 1+3442T4+p4T8 1 + 3442 T^{4} + p^{4} T^{8}
61C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
67C22C_2^2 (1−10T+50T2−10pT3+p2T4)2 ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
71C23C_2^3 1+5794T4+p4T8 1 + 5794 T^{4} + p^{4} T^{8}
73C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1+10T+pT2)4 ( 1 + 10 T + p T^{2} )^{4}
83C23C_2^3 1−3374T4+p4T8 1 - 3374 T^{4} + p^{4} T^{8}
89C23C_2^3 1−15518T4+p4T8 1 - 15518 T^{4} + p^{4} T^{8}
97C22C_2^2 (1+14T+98T2+14pT3+p2T4)2 ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{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

−7.22962409609456433245768818249, −7.07016635259605825682806147076, −6.86915256857119081157305746333, −6.55671272987272668850243569332, −6.28587054128424301410234678989, −6.11303450444885643996722076195, −5.64641845260248698870777580206, −5.58271078423323100671504545178, −5.51681358829249898529448746781, −5.26020901751273711743845626171, −4.87486010028116000615906200460, −4.30153597556576245932011039173, −4.10649225898848003879516446271, −3.77028440778335171036997422539, −3.73149944900423468378206603820, −3.54352803548533372590464931605, −3.42587723280838639040613155157, −3.10134724492919741928496646964, −2.61279009361964217256241870082, −2.51197808784028607522587919343, −2.35446875000867515281987056864, −1.69219593981253661912035641300, −1.55626953031201905766173472793, −1.15692906186722819687092818764, −0.29505375134976694326419730739, 0.29505375134976694326419730739, 1.15692906186722819687092818764, 1.55626953031201905766173472793, 1.69219593981253661912035641300, 2.35446875000867515281987056864, 2.51197808784028607522587919343, 2.61279009361964217256241870082, 3.10134724492919741928496646964, 3.42587723280838639040613155157, 3.54352803548533372590464931605, 3.73149944900423468378206603820, 3.77028440778335171036997422539, 4.10649225898848003879516446271, 4.30153597556576245932011039173, 4.87486010028116000615906200460, 5.26020901751273711743845626171, 5.51681358829249898529448746781, 5.58271078423323100671504545178, 5.64641845260248698870777580206, 6.11303450444885643996722076195, 6.28587054128424301410234678989, 6.55671272987272668850243569332, 6.86915256857119081157305746333, 7.07016635259605825682806147076, 7.22962409609456433245768818249

Graph of the ZZ-function along the critical line