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Properties

Label 4-800e2-1.1-c1e2-0-13
Degree 44
Conductor 640000640000
Sign 11
Analytic cond. 40.806940.8069
Root an. cond. 2.527452.52745
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  − 4·3-s − 4·7-s + 8·9-s + 2·13-s + 10·17-s + 8·19-s + 16·21-s − 4·23-s − 12·27-s − 2·37-s − 8·39-s + 12·43-s + 4·47-s + 8·49-s − 40·51-s + 14·53-s − 32·57-s + 8·59-s − 8·61-s − 32·63-s + 20·67-s + 16·69-s + 6·73-s + 32·79-s + 23·81-s + 4·83-s − 8·91-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.51·7-s + 8/3·9-s + 0.554·13-s + 2.42·17-s + 1.83·19-s + 3.49·21-s − 0.834·23-s − 2.30·27-s − 0.328·37-s − 1.28·39-s + 1.82·43-s + 0.583·47-s + 8/7·49-s − 5.60·51-s + 1.92·53-s − 4.23·57-s + 1.04·59-s − 1.02·61-s − 4.03·63-s + 2.44·67-s + 1.92·69-s + 0.702·73-s + 3.60·79-s + 23/9·81-s + 0.439·83-s − 0.838·91-s + ⋯

Functional equation

Λ(s)=(640000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(640000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 640000640000    =    210⋅542^{10} \cdot 5^{4}
Sign: 11
Analytic conductor: 40.806940.8069
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 640000, ( :1/2,1/2), 1)(4,\ 640000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.83170992910.8317099291
L(12)L(\frac12) ≈\approx 0.83170992910.8317099291
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
5 1 1
good3C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 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 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
37C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1−20T+200T2−20pT3+p2T4 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
73C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + 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

−10.29261216253602200569395796978, −10.29114244288198554202836159454, −9.751561531843854565132887073834, −9.465245993447043182873358665702, −9.056792251200399634530463683178, −8.206740596615492220000420459678, −7.56452075924571639174983515610, −7.54254681243446590685683358450, −6.77680705173972987379947169767, −6.44681744869604256638195753490, −5.94787183488346952596562801812, −5.69385524670392918644676982756, −5.28655135622173376336904391238, −5.08995377658841635889391485991, −3.98157943307833483926616460479, −3.66404240530070806974758690128, −3.22293548106315928613307069977, −2.26682324829930236713447204304, −0.875958458013331180292750983529, −0.798051517344053497583917184647, 0.798051517344053497583917184647, 0.875958458013331180292750983529, 2.26682324829930236713447204304, 3.22293548106315928613307069977, 3.66404240530070806974758690128, 3.98157943307833483926616460479, 5.08995377658841635889391485991, 5.28655135622173376336904391238, 5.69385524670392918644676982756, 5.94787183488346952596562801812, 6.44681744869604256638195753490, 6.77680705173972987379947169767, 7.54254681243446590685683358450, 7.56452075924571639174983515610, 8.206740596615492220000420459678, 9.056792251200399634530463683178, 9.465245993447043182873358665702, 9.751561531843854565132887073834, 10.29114244288198554202836159454, 10.29261216253602200569395796978

Graph of the ZZ-function along the critical line