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Properties

Label 4-160e2-1.1-c3e2-0-7
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 89.119389.1193
Root an. cond. 3.072503.07250
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·5-s − 34·9-s − 124·13-s − 92·17-s + 75·25-s − 180·29-s − 428·37-s − 20·41-s + 340·45-s + 294·49-s − 1.35e3·53-s + 500·61-s + 1.24e3·65-s + 1.04e3·73-s + 427·81-s + 920·85-s + 1.94e3·89-s − 1.86e3·97-s − 1.20e3·101-s + 4.30e3·109-s − 4.36e3·113-s + 4.21e3·117-s − 2.58e3·121-s − 500·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.25·9-s − 2.64·13-s − 1.31·17-s + 3/5·25-s − 1.15·29-s − 1.90·37-s − 0.0761·41-s + 1.12·45-s + 6/7·49-s − 3.51·53-s + 1.04·61-s + 2.36·65-s + 1.67·73-s + 0.585·81-s + 1.17·85-s + 2.31·89-s − 1.95·97-s − 1.18·101-s + 3.78·109-s − 3.63·113-s + 3.33·117-s − 1.93·121-s − 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 89.119389.1193
Root analytic conductor: 3.072503.07250
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 25600, ( :3/2,3/2), 1)(4,\ 25600,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1+34T2+p6T4 1 + 34 T^{2} + p^{6} T^{4}
7C22C_2^2 1−6p2T2+p6T4 1 - 6 p^{2} T^{2} + p^{6} T^{4}
11C22C_2^2 1+2582T2+p6T4 1 + 2582 T^{2} + p^{6} T^{4}
13C2C_2 (1+62T+p3T2)2 ( 1 + 62 T + p^{3} T^{2} )^{2}
17C2C_2 (1+46T+p3T2)2 ( 1 + 46 T + p^{3} T^{2} )^{2}
19C22C_2^2 1+2198T2+p6T4 1 + 2198 T^{2} + p^{6} T^{4}
23C22C_2^2 1−12646T2+p6T4 1 - 12646 T^{2} + p^{6} T^{4}
29C2C_2 (1+90T+p3T2)2 ( 1 + 90 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+36462T2+p6T4 1 + 36462 T^{2} + p^{6} T^{4}
37C2C_2 (1+214T+p3T2)2 ( 1 + 214 T + p^{3} T^{2} )^{2}
41C2C_2 (1+10T+p3T2)2 ( 1 + 10 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+154514T2+p6T4 1 + 154514 T^{2} + p^{6} T^{4}
47C22C_2^2 1+49226T2+p6T4 1 + 49226 T^{2} + p^{6} T^{4}
53C2C_2 (1+678T+p3T2)2 ( 1 + 678 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+241478T2+p6T4 1 + 241478 T^{2} + p^{6} T^{4}
61C2C_2 (1−250T+p3T2)2 ( 1 - 250 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+599106T2+p6T4 1 + 599106 T^{2} + p^{6} T^{4}
71C22C_2^2 1+581342T2+p6T4 1 + 581342 T^{2} + p^{6} T^{4}
73C2C_2 (1−522T+p3T2)2 ( 1 - 522 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+217758T2+p6T4 1 + 217758 T^{2} + p^{6} T^{4}
83C22C_2^2 1+999074T2+p6T4 1 + 999074 T^{2} + p^{6} T^{4}
89C2C_2 (1−970T+p3T2)2 ( 1 - 970 T + p^{3} T^{2} )^{2}
97C2C_2 (1+934T+p3T2)2 ( 1 + 934 T + p^{3} T^{2} )^{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

−12.14417407189950667783235657394, −11.72635719987079053075994375186, −11.19990040512809189244998901798, −10.85092715792755929576333917388, −10.18354381072050069418901439625, −9.506101788907985375513364111873, −9.120904275493744894244518951720, −8.590225515539206919452026639226, −7.80295355920318112079605761779, −7.64245355927736891419684440360, −6.82808572228664515094071631921, −6.47073839427840834831047605747, −5.25780113204589902706074526859, −5.14506212575153012544161861903, −4.34505219403782920223278302616, −3.50053540682691292918108792216, −2.71009757244035338446943905309, −2.05225850346205488929694359608, 0, 0, 2.05225850346205488929694359608, 2.71009757244035338446943905309, 3.50053540682691292918108792216, 4.34505219403782920223278302616, 5.14506212575153012544161861903, 5.25780113204589902706074526859, 6.47073839427840834831047605747, 6.82808572228664515094071631921, 7.64245355927736891419684440360, 7.80295355920318112079605761779, 8.590225515539206919452026639226, 9.120904275493744894244518951720, 9.506101788907985375513364111873, 10.18354381072050069418901439625, 10.85092715792755929576333917388, 11.19990040512809189244998901798, 11.72635719987079053075994375186, 12.14417407189950667783235657394

Graph of the ZZ-function along the critical line