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Properties

Label 8-960e4-1.1-c1e4-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 2·9-s − 16·23-s − 2·25-s − 16·31-s − 8·41-s + 16·47-s + 12·49-s + 16·63-s + 40·73-s − 16·79-s + 3·81-s − 24·89-s − 24·97-s − 8·103-s + 16·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.02·7-s − 2/3·9-s − 3.33·23-s − 2/5·25-s − 2.87·31-s − 1.24·41-s + 2.33·47-s + 12/7·49-s + 2.01·63-s + 4.68·73-s − 1.80·79-s + 1/3·81-s − 2.54·89-s − 2.43·97-s − 0.788·103-s + 1.50·113-s + 1.09·121-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 + 10.0·161-s + 0.0783·163-s + 0.0773·167-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 0.21546721660.2154672166
L(12)L(\frac12) ≈\approx 0.21546721660.2154672166
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+T2)2 ( 1 + T^{2} )^{2}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
11D4×C2D_4\times C_2 1−12T2+86T4−12p2T6+p4T8 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−20T2+246T4−20p2T6+p4T8 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1+22T2+p2T4)2 ( 1 + 22 T^{2} + p^{2} T^{4} )^{2}
19C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
23C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
29D4×C2D_4\times C_2 1−12T2+950T4−12p2T6+p4T8 1 - 12 T^{2} + 950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8}
31D4D_{4} (1+8T+66T2+8pT3+p2T4)2 ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−52T2+1686T4−52p2T6+p4T8 1 - 52 T^{2} + 1686 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1+4T+38T2+4pT3+p2T4)2 ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
53D4×C2D_4\times C_2 1−44T2−810T4−44p2T6+p4T8 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
59D4×C2D_4\times C_2 1−140T2+10134T4−140p2T6+p4T8 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
61C22C_2^2 (1−106T2+p2T4)2 ( 1 - 106 T^{2} + p^{2} T^{4} )^{2}
67C22C_2^2 (1−86T2+p2T4)2 ( 1 - 86 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
79D4D_{4} (1+8T+66T2+8pT3+p2T4)2 ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−102T2+p2T4)2 ( 1 - 102 T^{2} + p^{2} T^{4} )^{2}
89D4D_{4} (1+12T+166T2+12pT3+p2T4)2 ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
97D4D_{4} (1+12T+182T2+12pT3+p2T4)2 ( 1 + 12 T + 182 T^{2} + 12 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

−6.99121605412110690812953763470, −6.88227941416571216392276642571, −6.85116923055588219210580640108, −6.49068351820525679154532060251, −6.21691590901304568780341913096, −6.01805275804041882373092162212, −6.00275596419484365203390871301, −5.62549663075719825389114024011, −5.45019366981643563032793904104, −5.20202905090805955754151396484, −5.12603774425515550222629191946, −4.37583482859781024001677135729, −4.16650753041499254463902367224, −4.11004329381752899349374828778, −3.69788109104502254868975505276, −3.63581330590336187451320717646, −3.33007010539779106782583254286, −3.06127327217362228335623237215, −2.82846252159331757408680160201, −2.45274542310608711658207058440, −2.06663045020158302442231631586, −1.86095555405810258373922402236, −1.50093436801183107640589598370, −0.51045255556737264298873622346, −0.18572842624847487243850789688, 0.18572842624847487243850789688, 0.51045255556737264298873622346, 1.50093436801183107640589598370, 1.86095555405810258373922402236, 2.06663045020158302442231631586, 2.45274542310608711658207058440, 2.82846252159331757408680160201, 3.06127327217362228335623237215, 3.33007010539779106782583254286, 3.63581330590336187451320717646, 3.69788109104502254868975505276, 4.11004329381752899349374828778, 4.16650753041499254463902367224, 4.37583482859781024001677135729, 5.12603774425515550222629191946, 5.20202905090805955754151396484, 5.45019366981643563032793904104, 5.62549663075719825389114024011, 6.00275596419484365203390871301, 6.01805275804041882373092162212, 6.21691590901304568780341913096, 6.49068351820525679154532060251, 6.85116923055588219210580640108, 6.88227941416571216392276642571, 6.99121605412110690812953763470

Graph of the ZZ-function along the critical line