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Properties

Label 4-4050e2-1.1-c1e2-0-18
Degree 44
Conductor 1640250016402500
Sign 11
Analytic cond. 1045.831045.83
Root an. cond. 5.686775.68677
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  − 4-s + 16-s + 14·19-s + 12·29-s − 20·31-s + 18·41-s + 10·49-s + 18·59-s − 8·61-s − 64-s + 12·71-s − 14·76-s − 4·79-s − 30·89-s + 24·101-s − 4·109-s − 12·116-s − 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s + 3.21·19-s + 2.22·29-s − 3.59·31-s + 2.81·41-s + 10/7·49-s + 2.34·59-s − 1.02·61-s − 1/8·64-s + 1.42·71-s − 1.60·76-s − 0.450·79-s − 3.17·89-s + 2.38·101-s − 0.383·109-s − 1.11·116-s − 2·121-s + 1.79·124-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 + ⋯

Functional equation

Λ(s)=(16402500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16402500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1640250016402500    =    22⋅38⋅542^{2} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1045.831045.83
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16402500, ( :1/2,1/2), 1)(4,\ 16402500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8643014522.864301452
L(12)L(\frac12) ≈\approx 2.8643014522.864301452
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)
bad2C2C_2 1+T2 1 + T^{2}
3 1 1
5 1 1
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
43C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
47C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1+35T2+p2T4 1 + 35 T^{2} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1−145T2+p2T4 1 - 145 T^{2} + p^{2} T^{4}
79C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
83C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
89C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
97C22C_2^2 1+95T2+p2T4 1 + 95 T^{2} + p^{2} T^{4}
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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

−8.805856177567859002070517205193, −8.238351007039741974527337009330, −7.69902753231507802638820379407, −7.57722258358808735598727363065, −7.31226451940302336730862685322, −6.86909677353774157660711830655, −6.50318940822930695702618271278, −5.70264484843409895762071121840, −5.69807111918314342837917991461, −5.28887201959665023928439923614, −5.07810895614880171917144638683, −4.34608489938324046090043032395, −4.09099822425226783812750964769, −3.60672087724269147412590314271, −3.25571159188896679213064215776, −2.70865202978629777466092573965, −2.38779048958888832617022952129, −1.53438325158638567969320821890, −1.04347743841135907048745303464, −0.59433246210078581894413624233, 0.59433246210078581894413624233, 1.04347743841135907048745303464, 1.53438325158638567969320821890, 2.38779048958888832617022952129, 2.70865202978629777466092573965, 3.25571159188896679213064215776, 3.60672087724269147412590314271, 4.09099822425226783812750964769, 4.34608489938324046090043032395, 5.07810895614880171917144638683, 5.28887201959665023928439923614, 5.69807111918314342837917991461, 5.70264484843409895762071121840, 6.50318940822930695702618271278, 6.86909677353774157660711830655, 7.31226451940302336730862685322, 7.57722258358808735598727363065, 7.69902753231507802638820379407, 8.238351007039741974527337009330, 8.805856177567859002070517205193

Graph of the ZZ-function along the critical line