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Properties

Label 4-2e16-1.1-c1e2-0-11
Degree 44
Conductor 6553665536
Sign −1-1
Analytic cond. 4.178634.17863
Root an. cond. 1.429741.42974
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·9-s − 12·17-s − 10·25-s + 12·41-s − 14·49-s − 4·73-s − 5·81-s − 36·89-s + 20·97-s + 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s − 2.91·17-s − 2·25-s + 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s − 3.81·89-s + 2.03·97-s + 3.38·113-s + 1.27·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-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)=(65536s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(65536s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6553665536    =    2162^{16}
Sign: −1-1
Analytic conductor: 4.178634.17863
Root analytic conductor: 1.429741.42974
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 65536, ( :1/2,1/2), −1)(4,\ 65536,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
89C2C_2 (1+18T+pT2)2 ( 1 + 18 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.739977345448053506810012798737, −9.084650424863502138877874475320, −8.559961689771162774439116698021, −8.337928305679105143801605101388, −7.49638267192940041493128754308, −7.14838005963241329022744903293, −6.29051795613585298614624812000, −6.13287781737238367476060833635, −5.43412469058090981748265870176, −4.43148007247498493089277458199, −4.38680180444175422137402538929, −3.38746738060370983088123271545, −2.49619173311123854201331757855, −1.90996633779170653420430529631, 0, 1.90996633779170653420430529631, 2.49619173311123854201331757855, 3.38746738060370983088123271545, 4.38680180444175422137402538929, 4.43148007247498493089277458199, 5.43412469058090981748265870176, 6.13287781737238367476060833635, 6.29051795613585298614624812000, 7.14838005963241329022744903293, 7.49638267192940041493128754308, 8.337928305679105143801605101388, 8.559961689771162774439116698021, 9.084650424863502138877874475320, 9.739977345448053506810012798737

Graph of the ZZ-function along the critical line