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Properties

Label 8-2160e4-1.1-c1e4-0-18
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·19-s + 9·25-s − 28·31-s − 10·49-s − 16·61-s − 40·109-s − 6·121-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 + 227-s + ⋯
L(s)  = 1  + 5.50·19-s + 9/5·25-s − 5.02·31-s − 1.42·49-s − 2.04·61-s − 3.83·109-s − 0.545·121-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 + 0.0663·227-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 4.9782045594.978204559
L(12)L(\frac12) ≈\approx 4.9782045594.978204559
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
5C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
good7C2C_2 (1−3T+pT2)2(1+3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2}
11C22C_2^2 (1+3T2+p2T4)2 ( 1 + 3 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17C22C_2^2 (1−18T2+p2T4)2 ( 1 - 18 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
23C22C_2^2 (1−42T2+p2T4)2 ( 1 - 42 T^{2} + p^{2} T^{4} )^{2}
29C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
31C2C_2 (1+7T+pT2)4 ( 1 + 7 T + p T^{2} )^{4}
37C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
41C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1−90T2+p2T4)2 ( 1 - 90 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−97T2+p2T4)2 ( 1 - 97 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+42T2+p2T4)2 ( 1 + 42 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
67C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C22C_2^2 (1−127T2+p2T4)2 ( 1 - 127 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C22C_2^2 (1−141T2+p2T4)2 ( 1 - 141 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+102T2+p2T4)2 ( 1 + 102 T^{2} + p^{2} T^{4} )^{2}
97C22C_2^2 (1−175T2+p2T4)2 ( 1 - 175 T^{2} + 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

−6.70400955932220577077470496976, −6.14760465331538496449936466691, −5.88486428964193287050253707565, −5.70442166840490137390538396995, −5.45132019166679165909671016355, −5.40009491695299035049301584719, −5.33965992042105433312300028155, −5.14456478140879043960075343130, −4.82326152430334220606221475055, −4.49990142408601185397172809092, −4.46896452847844228530078010632, −3.91419685220581800678477012977, −3.81824299601216125343276134828, −3.40413890904809460604931686374, −3.37656389073463306410437775450, −3.20200839323780844130701654935, −2.99464306180405638829342923747, −2.72344532496388764028809155494, −2.48755056635725276733944794371, −1.83559711184905792660262808855, −1.56666601207922982139473079595, −1.54061628817866170653356116432, −1.29185300653474010907580010010, −0.55894289828552607051445944650, −0.54005133197055822449885771424, 0.54005133197055822449885771424, 0.55894289828552607051445944650, 1.29185300653474010907580010010, 1.54061628817866170653356116432, 1.56666601207922982139473079595, 1.83559711184905792660262808855, 2.48755056635725276733944794371, 2.72344532496388764028809155494, 2.99464306180405638829342923747, 3.20200839323780844130701654935, 3.37656389073463306410437775450, 3.40413890904809460604931686374, 3.81824299601216125343276134828, 3.91419685220581800678477012977, 4.46896452847844228530078010632, 4.49990142408601185397172809092, 4.82326152430334220606221475055, 5.14456478140879043960075343130, 5.33965992042105433312300028155, 5.40009491695299035049301584719, 5.45132019166679165909671016355, 5.70442166840490137390538396995, 5.88486428964193287050253707565, 6.14760465331538496449936466691, 6.70400955932220577077470496976

Graph of the ZZ-function along the critical line