↑

Properties

Label 4-600e2-1.1-c1e2-0-3
Degree 44
Conductor 360000360000
Sign 11
Analytic cond. 22.953922.9539
Root an. cond. 2.188842.18884
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·9-s + 2·12-s − 4·16-s + 4·18-s − 5·27-s + 16·31-s + 8·32-s − 4·36-s − 4·48-s − 10·49-s − 8·53-s + 10·54-s − 32·62-s − 8·64-s + 20·79-s + 81-s + 18·83-s + 16·93-s + 8·96-s + 20·98-s + 16·106-s − 34·107-s − 10·108-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 2/3·9-s + 0.577·12-s − 16-s + 0.942·18-s − 0.962·27-s + 2.87·31-s + 1.41·32-s − 2/3·36-s − 0.577·48-s − 1.42·49-s − 1.09·53-s + 1.36·54-s − 4.06·62-s − 64-s + 2.25·79-s + 1/9·81-s + 1.97·83-s + 1.65·93-s + 0.816·96-s + 2.02·98-s + 1.55·106-s − 3.28·107-s − 0.962·108-s + ⋯

Functional equation

Λ(s)=(360000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(360000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 360000360000    =    26⋅32⋅542^{6} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 22.953922.9539
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 360000, ( :1/2,1/2), 1)(4,\ 360000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.85924172780.8592417278
L(12)L(\frac12) ≈\approx 0.85924172780.8592417278
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+pT+pT2 1 + p T + p T^{2}
3C2C_2 1−T+pT2 1 - T + p T^{2}
5 1 1
good7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
71C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
73C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
79C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
83C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
89C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
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

−8.665421196412280182170334766998, −8.182910838086186172188996077240, −7.974950820166071766849701202962, −7.77921455567468453333623630561, −6.88699879047458142537014880578, −6.54282764581716256623518288017, −6.19421351777184295505297292819, −5.39258018193498154600079656616, −4.80562532332567230387554534751, −4.35480286006529544992633071822, −3.52151660613402712389144369009, −2.90862550946813728509296480615, −2.38035753805067884592817569239, −1.61243686764748739595327097028, −0.64659822666269902998848780560, 0.64659822666269902998848780560, 1.61243686764748739595327097028, 2.38035753805067884592817569239, 2.90862550946813728509296480615, 3.52151660613402712389144369009, 4.35480286006529544992633071822, 4.80562532332567230387554534751, 5.39258018193498154600079656616, 6.19421351777184295505297292819, 6.54282764581716256623518288017, 6.88699879047458142537014880578, 7.77921455567468453333623630561, 7.974950820166071766849701202962, 8.182910838086186172188996077240, 8.665421196412280182170334766998

Graph of the ZZ-function along the critical line