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Properties

Label 4-4050e2-1.1-c1e2-0-9
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 12·11-s + 16-s − 4·19-s + 16·31-s − 6·41-s + 12·44-s + 13·49-s + 12·59-s − 20·61-s − 64-s − 24·71-s + 4·76-s + 26·79-s + 18·89-s − 12·101-s + 32·109-s + 86·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s − 3.61·11-s + 1/4·16-s − 0.917·19-s + 2.87·31-s − 0.937·41-s + 1.80·44-s + 13/7·49-s + 1.56·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 0.458·76-s + 2.92·79-s + 1.90·89-s − 1.19·101-s + 3.06·109-s + 7.81·121-s − 1.43·124-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 0.87503639170.8750363917
L(12)L(\frac12) ≈\approx 0.87503639170.8750363917
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 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 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+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
43C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
47C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
79C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1−9T+pT2)2 ( 1 - 9 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

−8.443543527109999486040773415037, −8.360513599788231649305551092323, −7.78861391901733425968005862456, −7.66664550777163495166842180272, −7.43454251005256896350504534642, −6.72223064171189385323875535870, −6.43968769814352673672335687940, −5.79981619360413301517388524391, −5.76711286000322917290163386923, −5.13452699826958485669962061430, −4.93109958198209828821033174466, −4.45990439335744863555543223553, −4.37840461121146647123840362926, −3.42651788310693317236485796022, −3.17253054456386988673960947603, −2.51259306415624349843873275129, −2.50947775301949639753157053890, −1.88648311828018344086146277706, −0.880704118397224503250104996619, −0.33230590500235985266084713897, 0.33230590500235985266084713897, 0.880704118397224503250104996619, 1.88648311828018344086146277706, 2.50947775301949639753157053890, 2.51259306415624349843873275129, 3.17253054456386988673960947603, 3.42651788310693317236485796022, 4.37840461121146647123840362926, 4.45990439335744863555543223553, 4.93109958198209828821033174466, 5.13452699826958485669962061430, 5.76711286000322917290163386923, 5.79981619360413301517388524391, 6.43968769814352673672335687940, 6.72223064171189385323875535870, 7.43454251005256896350504534642, 7.66664550777163495166842180272, 7.78861391901733425968005862456, 8.360513599788231649305551092323, 8.443543527109999486040773415037

Graph of the ZZ-function along the critical line