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Properties

Label 4-128000-1.1-c1e2-0-4
Degree 44
Conductor 128000128000
Sign 11
Analytic cond. 8.161398.16139
Root an. cond. 1.690211.69021
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 − 5-s + 6·9-s + 4·13-s − 4·15-s + 25-s − 4·27-s + 8·31-s + 4·37-s + 16·39-s + 12·41-s + 20·43-s − 6·45-s − 10·49-s − 12·53-s − 4·65-s − 4·67-s + 24·71-s + 4·75-s − 16·79-s − 37·81-s − 12·83-s − 12·89-s + 32·93-s + 12·107-s + 16·111-s + 24·117-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.447·5-s + 2·9-s + 1.10·13-s − 1.03·15-s + 1/5·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 2.56·39-s + 1.87·41-s + 3.04·43-s − 0.894·45-s − 1.42·49-s − 1.64·53-s − 0.496·65-s − 0.488·67-s + 2.84·71-s + 0.461·75-s − 1.80·79-s − 4.11·81-s − 1.31·83-s − 1.27·89-s + 3.31·93-s + 1.16·107-s + 1.51·111-s + 2.21·117-s + ⋯

Functional equation

Λ(s)=(128000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(128000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 128000128000    =    210⋅532^{10} \cdot 5^{3}
Sign: 11
Analytic conductor: 8.161398.16139
Root analytic conductor: 1.690211.69021
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 128000, ( :1/2,1/2), 1)(4,\ 128000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2709508193.270950819
L(12)L(\frac12) ≈\approx 3.2709508193.270950819
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
5C1C_1 1+T 1 + T
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−9.249279365224539237566485048625, −8.740293324354028093320034492248, −8.550149092441660717391493732033, −7.950176007435402201982724906846, −7.71273110823499542846308181544, −7.33279621647751156873215628670, −6.27087624192875571051851265258, −6.14181160684855424693200797613, −5.25974231276140424950498810771, −4.19097847493160731891690631788, −4.12250433368686324236236368171, −3.32887958847832114233485788826, −2.76929890617261215013507568311, −2.41662881718811067600819256762, −1.26575059491370931038306222328, 1.26575059491370931038306222328, 2.41662881718811067600819256762, 2.76929890617261215013507568311, 3.32887958847832114233485788826, 4.12250433368686324236236368171, 4.19097847493160731891690631788, 5.25974231276140424950498810771, 6.14181160684855424693200797613, 6.27087624192875571051851265258, 7.33279621647751156873215628670, 7.71273110823499542846308181544, 7.950176007435402201982724906846, 8.550149092441660717391493732033, 8.740293324354028093320034492248, 9.249279365224539237566485048625

Graph of the ZZ-function along the critical line