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Properties

Label 4-1280e2-1.1-c1e2-0-15
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·13-s + 6·17-s − 25-s − 8·29-s − 14·37-s + 16·41-s − 18·53-s − 4·65-s + 22·73-s − 9·81-s + 12·85-s + 26·97-s + 12·109-s − 2·113-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.554·13-s + 1.45·17-s − 1/5·25-s − 1.48·29-s − 2.30·37-s + 2.49·41-s − 2.47·53-s − 0.496·65-s + 2.57·73-s − 81-s + 1.30·85-s + 2.63·97-s + 1.14·109-s − 0.188·113-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2988569302.298856930
L(12)L(\frac12) ≈\approx 2.2988569302.298856930
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
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
7C22C_2^2 1+p2T4 1 + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
23C22C_2^2 1+p2T4 1 + p^{2} T^{4}
29C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1+2T+pT2)(1+12T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
43C22C_2^2 1+p2T4 1 + p^{2} T^{4}
47C22C_2^2 1+p2T4 1 + p^{2} T^{4}
53C2C_2 (1+4T+pT2)(1+14T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1+p2T4 1 + p^{2} T^{4}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1−6T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+p2T4 1 + p^{2} T^{4}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1−8T+pT2) ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{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

−10.05879794562371001412071309452, −9.382472361501381457618097390178, −9.219060541222198231611306318034, −8.863510508636534688063042677467, −8.062749379156499831713267310079, −7.83146047766568019645839035004, −7.55804970294160445959981322978, −6.97380082819873646303314417643, −6.58776436759169355051601709330, −5.98158107861579876239433546278, −5.74677689663861871884237201582, −5.27988271403578315098341106402, −4.95350834815705090788492125697, −4.30204232496745581832036766413, −3.68735195784165874322770106795, −3.29380686358137115966435644102, −2.71012991301328914697046270844, −1.92679274233205558666457369245, −1.68741738297018311505063943572, −0.63131316802742449880273092907, 0.63131316802742449880273092907, 1.68741738297018311505063943572, 1.92679274233205558666457369245, 2.71012991301328914697046270844, 3.29380686358137115966435644102, 3.68735195784165874322770106795, 4.30204232496745581832036766413, 4.95350834815705090788492125697, 5.27988271403578315098341106402, 5.74677689663861871884237201582, 5.98158107861579876239433546278, 6.58776436759169355051601709330, 6.97380082819873646303314417643, 7.55804970294160445959981322978, 7.83146047766568019645839035004, 8.062749379156499831713267310079, 8.863510508636534688063042677467, 9.219060541222198231611306318034, 9.382472361501381457618097390178, 10.05879794562371001412071309452

Graph of the ZZ-function along the critical line