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Properties

Label 4-700e2-1.1-c4e2-0-0
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 5235.825235.82
Root an. cond. 8.506408.50640
Motivic weight 44
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  + 14·7-s + 114·9-s + 36·11-s − 1.47e3·23-s − 1.69e3·29-s − 4.77e3·37-s + 5.02e3·43-s − 2.20e3·49-s + 540·53-s + 1.59e3·63-s − 4.90e3·67-s − 6.30e3·71-s + 504·77-s − 7.96e3·79-s + 6.43e3·81-s + 4.10e3·99-s + 2.58e4·107-s − 1.40e4·109-s − 3.74e4·113-s − 2.83e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2/7·7-s + 1.40·9-s + 0.297·11-s − 2.79·23-s − 2.01·29-s − 3.48·37-s + 2.71·43-s − 0.918·49-s + 0.192·53-s + 0.402·63-s − 1.09·67-s − 1.24·71-s + 0.0850·77-s − 1.27·79-s + 0.980·81-s + 0.418·99-s + 2.26·107-s − 1.18·109-s − 2.93·113-s − 1.93·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 5235.825235.82
Root analytic conductor: 8.506408.50640
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :2,2), 1)(4,\ 490000,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.87613835310.8761383531
L(12)L(\frac12) ≈\approx 0.87613835310.8761383531
L(3)L(3) 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
7C2C_2 1−2pT+p4T2 1 - 2 p T + p^{4} T^{2}
good3C22C_2^2 1−38pT2+p8T4 1 - 38 p T^{2} + p^{8} T^{4}
11C2C_2 (1−18T+p4T2)2 ( 1 - 18 T + p^{4} T^{2} )^{2}
13C22C_2^2 1−39794T2+p8T4 1 - 39794 T^{2} + p^{8} T^{4}
17C22C_2^2 1+5758T2+p8T4 1 + 5758 T^{2} + p^{8} T^{4}
19C22C_2^2 1−252530T2+p8T4 1 - 252530 T^{2} + p^{8} T^{4}
23C2C_2 (1+738T+p4T2)2 ( 1 + 738 T + p^{4} T^{2} )^{2}
29C2C_2 (1+846T+p4T2)2 ( 1 + 846 T + p^{4} T^{2} )^{2}
31C22C_2^2 1−492290T2+p8T4 1 - 492290 T^{2} + p^{8} T^{4}
37C2C_2 (1+2386T+p4T2)2 ( 1 + 2386 T + p^{4} T^{2} )^{2}
41C22C_2^2 1−1634p2T2+p8T4 1 - 1634 p^{2} T^{2} + p^{8} T^{4}
43C2C_2 (1−2510T+p4T2)2 ( 1 - 2510 T + p^{4} T^{2} )^{2}
47C22C_2^2 1+1859710T2+p8T4 1 + 1859710 T^{2} + p^{8} T^{4}
53C2C_2 (1−270T+p4T2)2 ( 1 - 270 T + p^{4} T^{2} )^{2}
59C22C_2^2 1−14384690T2+p8T4 1 - 14384690 T^{2} + p^{8} T^{4}
61C22C_2^2 1+14450830T2+p8T4 1 + 14450830 T^{2} + p^{8} T^{4}
67C2C_2 (1+2450T+p4T2)2 ( 1 + 2450 T + p^{4} T^{2} )^{2}
71C2C_2 (1+3150T+p4T2)2 ( 1 + 3150 T + p^{4} T^{2} )^{2}
73C22C_2^2 1−56740994T2+p8T4 1 - 56740994 T^{2} + p^{8} T^{4}
79C2C_2 (1+3982T+p4T2)2 ( 1 + 3982 T + p^{4} T^{2} )^{2}
83C22C_2^2 1−69825650T2+p8T4 1 - 69825650 T^{2} + p^{8} T^{4}
89C22C_2^2 1−67615490T2+p8T4 1 - 67615490 T^{2} + p^{8} T^{4}
97C22C_2^2 1−18761474T2+p8T4 1 - 18761474 T^{2} + p^{8} 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

−10.27167720455696130526906975223, −9.627951553464494981825681262773, −9.142620243451089958744362312243, −9.040611583774664931166516435179, −8.135345277173917889542950938711, −8.004175492190739363139963866269, −7.38518943511079337723985823732, −7.10152686186682762229001942178, −6.69577614957683683232954837513, −5.83392955774283053011651104565, −5.80244210336991880678015217983, −5.15221363714083536214505316019, −4.33652625458475920103268608635, −4.15675712772178906034254704347, −3.72644517934444536430403603759, −3.06608341267545561159244261436, −1.94486211140506949202479301456, −1.91608973385211524446509398848, −1.28385552421691757621432388207, −0.21604643499578151548730473781, 0.21604643499578151548730473781, 1.28385552421691757621432388207, 1.91608973385211524446509398848, 1.94486211140506949202479301456, 3.06608341267545561159244261436, 3.72644517934444536430403603759, 4.15675712772178906034254704347, 4.33652625458475920103268608635, 5.15221363714083536214505316019, 5.80244210336991880678015217983, 5.83392955774283053011651104565, 6.69577614957683683232954837513, 7.10152686186682762229001942178, 7.38518943511079337723985823732, 8.004175492190739363139963866269, 8.135345277173917889542950938711, 9.040611583774664931166516435179, 9.142620243451089958744362312243, 9.627951553464494981825681262773, 10.27167720455696130526906975223

Graph of the ZZ-function along the critical line