↑

Properties

Label 4-950e2-1.1-c1e2-0-6
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 57.544157.5441
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5·9-s + 16-s − 2·19-s + 6·29-s + 4·31-s − 5·36-s + 12·41-s + 13·49-s − 6·59-s + 16·61-s − 64-s + 24·71-s + 2·76-s − 28·79-s + 16·81-s − 12·89-s + 24·101-s − 22·109-s − 6·116-s − 22·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1/4·16-s − 0.458·19-s + 1.11·29-s + 0.718·31-s − 5/6·36-s + 1.87·41-s + 13/7·49-s − 0.781·59-s + 2.04·61-s − 1/8·64-s + 2.84·71-s + 0.229·76-s − 3.15·79-s + 16/9·81-s − 1.27·89-s + 2.38·101-s − 2.10·109-s − 0.557·116-s − 2·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 57.544157.5441
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :1/2,1/2), 1)(4,\ 902500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2488119552.248811955
L(12)L(\frac12) ≈\approx 2.2488119552.248811955
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}
5 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
23C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
59C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C22C_2^2 1−85T2+p2T4 1 - 85 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+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
show more
show less
   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

−10.14412570831628558915944559515, −9.871140753453987620504090159995, −9.499938893504928702570356864974, −9.022994393690980526882549759416, −8.632754664683448521621364142105, −8.112494231222844401996753182294, −7.80253861516502004887216369859, −7.27299756019718214529961900449, −6.79808793499429898924222457416, −6.62009187741297926155647777340, −5.84379151744724813970441112931, −5.55267944731353413432100877937, −4.71600772933333300382195326786, −4.60404974447699729840986976376, −3.90831902467415936018583668795, −3.80258792603032100861754452958, −2.71880443125033685463285670275, −2.34385773632888722896866027426, −1.37013021504630416699966885410, −0.803118169931028016795268971038, 0.803118169931028016795268971038, 1.37013021504630416699966885410, 2.34385773632888722896866027426, 2.71880443125033685463285670275, 3.80258792603032100861754452958, 3.90831902467415936018583668795, 4.60404974447699729840986976376, 4.71600772933333300382195326786, 5.55267944731353413432100877937, 5.84379151744724813970441112931, 6.62009187741297926155647777340, 6.79808793499429898924222457416, 7.27299756019718214529961900449, 7.80253861516502004887216369859, 8.112494231222844401996753182294, 8.632754664683448521621364142105, 9.022994393690980526882549759416, 9.499938893504928702570356864974, 9.871140753453987620504090159995, 10.14412570831628558915944559515

Graph of the ZZ-function along the critical line