↑

Properties

Label 4-3e10-1.1-c1e2-0-2
Degree 44
Conductor 5904959049
Sign 11
Analytic cond. 3.765013.76501
Root an. cond. 1.392961.39296
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  + 2·4-s + 4·7-s − 2·13-s − 2·19-s − 4·25-s + 8·28-s − 2·31-s + 16·37-s + 22·43-s − 2·49-s − 4·52-s + 10·61-s − 8·64-s − 14·67-s + 22·73-s − 4·76-s − 14·79-s − 8·91-s − 14·97-s − 8·100-s − 14·103-s − 2·109-s − 16·121-s − 4·124-s + 127-s + 131-s − 8·133-s + ⋯
L(s)  = 1  + 4-s + 1.51·7-s − 0.554·13-s − 0.458·19-s − 4/5·25-s + 1.51·28-s − 0.359·31-s + 2.63·37-s + 3.35·43-s − 2/7·49-s − 0.554·52-s + 1.28·61-s − 64-s − 1.71·67-s + 2.57·73-s − 0.458·76-s − 1.57·79-s − 0.838·91-s − 1.42·97-s − 4/5·100-s − 1.37·103-s − 0.191·109-s − 1.45·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯

Functional equation

Λ(s)=(59049s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(59049s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5904959049    =    3103^{10}
Sign: 11
Analytic conductor: 3.765013.76501
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 59049, ( :1/2,1/2), 1)(4,\ 59049,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0184725832.018472583
L(12)L(\frac12) ≈\approx 2.0184725832.018472583
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)
bad3 1 1
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
5C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
13C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
17C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
29C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
31C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
43C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
47C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
53C22C_2^2 1+52T2+p2T4 1 + 52 T^{2} + p^{2} T^{4}
59C22C_2^2 1+112T2+p2T4 1 + 112 T^{2} + p^{2} T^{4}
61C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
67C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
71C22C_2^2 1+88T2+p2T4 1 + 88 T^{2} + p^{2} T^{4}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
83C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{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

−12.18863131069501005241351840304, −11.71630489752962138017706271769, −11.36341435559568201719934172804, −10.98957259426085812164359937729, −10.72633054857364640549729516398, −10.01624557525827670252698459237, −9.373503120959489145087639531418, −9.086887255792082165042347510051, −8.195021880370970742472551747703, −7.79886961415182704033617479477, −7.60162825894856137206908174096, −6.88522256932747131908927626690, −6.28792838733343724695047154582, −5.74897891895457914005886933698, −5.14855857250750191167305147161, −4.35591997396063863904045940049, −4.05301410065910786040085000782, −2.65079034130041710876826775177, −2.34752347581144079652748859470, −1.35764139016953185842768122715, 1.35764139016953185842768122715, 2.34752347581144079652748859470, 2.65079034130041710876826775177, 4.05301410065910786040085000782, 4.35591997396063863904045940049, 5.14855857250750191167305147161, 5.74897891895457914005886933698, 6.28792838733343724695047154582, 6.88522256932747131908927626690, 7.60162825894856137206908174096, 7.79886961415182704033617479477, 8.195021880370970742472551747703, 9.086887255792082165042347510051, 9.373503120959489145087639531418, 10.01624557525827670252698459237, 10.72633054857364640549729516398, 10.98957259426085812164359937729, 11.36341435559568201719934172804, 11.71630489752962138017706271769, 12.18863131069501005241351840304

Graph of the ZZ-function along the critical line