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Properties

Label 4-700e2-1.1-c3e2-0-5
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 1705.801705.80
Root an. cond. 6.426616.42661
Motivic weight 33
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  − 10·9-s + 56·11-s − 16·19-s − 348·29-s − 304·31-s + 100·41-s − 49·49-s + 544·59-s − 1.32e3·61-s − 1.76e3·71-s + 1.20e3·79-s − 629·81-s − 1.39e3·89-s − 560·99-s + 708·101-s + 292·109-s − 310·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.33e3·169-s + ⋯
L(s)  = 1  − 0.370·9-s + 1.53·11-s − 0.193·19-s − 2.22·29-s − 1.76·31-s + 0.380·41-s − 1/7·49-s + 1.20·59-s − 2.77·61-s − 2.94·71-s + 1.70·79-s − 0.862·81-s − 1.66·89-s − 0.568·99-s + 0.697·101-s + 0.256·109-s − 0.232·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.06·169-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{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: 1705.801705.80
Root analytic conductor: 6.426616.42661
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :3/2,3/2), 1)(4,\ 490000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2181952511.218195251
L(12)L(\frac12) ≈\approx 1.2181952511.218195251
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
5 1 1
7C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1+10T2+p6T4 1 + 10 T^{2} + p^{6} T^{4}
11C2C_2 (1−28T+p3T2)2 ( 1 - 28 T + p^{3} T^{2} )^{2}
13C22C_2^2 1+2330T2+p6T4 1 + 2330 T^{2} + p^{6} T^{4}
17C22C_2^2 1−7710T2+p6T4 1 - 7710 T^{2} + p^{6} T^{4}
19C2C_2 (1+8T+p3T2)2 ( 1 + 8 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−7950T2+p6T4 1 - 7950 T^{2} + p^{6} T^{4}
29C2C_2 (1+6pT+p3T2)2 ( 1 + 6 p T + p^{3} T^{2} )^{2}
31C2C_2 (1+152T+p3T2)2 ( 1 + 152 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−17206T2+p6T4 1 - 17206 T^{2} + p^{6} T^{4}
41C2C_2 (1−50T+p3T2)2 ( 1 - 50 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−2198T2+p6T4 1 - 2198 T^{2} + p^{6} T^{4}
47C22C_2^2 1−120030T2+p6T4 1 - 120030 T^{2} + p^{6} T^{4}
53C22C_2^2 1+27146T2+p6T4 1 + 27146 T^{2} + p^{6} T^{4}
59C2C_2 (1−272T+p3T2)2 ( 1 - 272 T + p^{3} T^{2} )^{2}
61C2C_2 (1+662T+p3T2)2 ( 1 + 662 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+165850T2+p6T4 1 + 165850 T^{2} + p^{6} T^{4}
71C2C_2 (1+880T+p3T2)2 ( 1 + 880 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−370990T2+p6T4 1 - 370990 T^{2} + p^{6} T^{4}
79C2C_2 (1−600T+p3T2)2 ( 1 - 600 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−754198T2+p6T4 1 - 754198 T^{2} + p^{6} T^{4}
89C2C_2 (1+698T+p3T2)2 ( 1 + 698 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1256830T2+p6T4 1 - 1256830 T^{2} + p^{6} 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

−10.55238199537600222012803758495, −9.499454328805588279025131180738, −9.348723809694728938475439172381, −9.228259159953760951262975320245, −8.572871489071158700617674233676, −8.214284632426944496340882235301, −7.47864079344163137652058518185, −7.24808176302326686493362470518, −6.87544597511449857560571261948, −6.07221787033952056893091077214, −5.90827927603145375737447094643, −5.46654608291612254616942162197, −4.70028353381260808266200760309, −4.23532740230581283573488280724, −3.67382707712152931985444751142, −3.35776654549675455023011525330, −2.53256220596016406188806003134, −1.72200907760468139473499418965, −1.42815544350459708405149936829, −0.30524584098653598233976479280, 0.30524584098653598233976479280, 1.42815544350459708405149936829, 1.72200907760468139473499418965, 2.53256220596016406188806003134, 3.35776654549675455023011525330, 3.67382707712152931985444751142, 4.23532740230581283573488280724, 4.70028353381260808266200760309, 5.46654608291612254616942162197, 5.90827927603145375737447094643, 6.07221787033952056893091077214, 6.87544597511449857560571261948, 7.24808176302326686493362470518, 7.47864079344163137652058518185, 8.214284632426944496340882235301, 8.572871489071158700617674233676, 9.228259159953760951262975320245, 9.348723809694728938475439172381, 9.499454328805588279025131180738, 10.55238199537600222012803758495

Graph of the ZZ-function along the critical line