Properties

Label 4-588e2-1.1-c5e2-0-13
Degree 44
Conductor 345744345744
Sign 11
Analytic cond. 8893.568893.56
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 18·3-s + 47·5-s + 243·9-s − 407·11-s + 449·13-s + 846·15-s − 1.86e3·17-s − 1.46e3·19-s − 44·23-s − 2.82e3·25-s + 2.91e3·27-s + 767·29-s − 1.11e4·31-s − 7.32e3·33-s − 3.11e3·37-s + 8.08e3·39-s − 7.84e3·41-s − 1.26e4·43-s + 1.14e4·45-s + 9.57e3·47-s − 3.36e4·51-s + 1.33e4·53-s − 1.91e4·55-s − 2.63e4·57-s − 4.75e4·59-s − 6.36e4·61-s + 2.11e4·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.840·5-s + 9-s − 1.01·11-s + 0.736·13-s + 0.970·15-s − 1.56·17-s − 0.929·19-s − 0.0173·23-s − 0.903·25-s + 0.769·27-s + 0.169·29-s − 2.08·31-s − 1.17·33-s − 0.373·37-s + 0.850·39-s − 0.728·41-s − 1.04·43-s + 0.840·45-s + 0.632·47-s − 1.81·51-s + 0.655·53-s − 0.852·55-s − 1.07·57-s − 1.77·59-s − 2.19·61-s + 0.619·65-s + ⋯

Functional equation

Λ(s)=(345744s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(345744s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345744345744    =    2432742^{4} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 8893.568893.56
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 345744, ( :5/2,5/2), 1)(4,\ 345744,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
7 1 1
good5D4D_{4} 147T+5032T247p5T3+p10T4 1 - 47 T + 5032 T^{2} - 47 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+37pT+361744T2+37p6T3+p10T4 1 + 37 p T + 361744 T^{2} + 37 p^{6} T^{3} + p^{10} T^{4}
13D4D_{4} 1449T+748730T2449p5T3+p10T4 1 - 449 T + 748730 T^{2} - 449 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+1868T+3683746T2+1868p5T3+p10T4 1 + 1868 T + 3683746 T^{2} + 1868 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1+77pT+882870T2+77p6T3+p10T4 1 + 77 p T + 882870 T^{2} + 77 p^{6} T^{3} + p^{10} T^{4}
23D4D_{4} 1+44T4829330T2+44p5T3+p10T4 1 + 44 T - 4829330 T^{2} + 44 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1767T+20137030T2767p5T3+p10T4 1 - 767 T + 20137030 T^{2} - 767 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+11170T+78225563T2+11170p5T3+p10T4 1 + 11170 T + 78225563 T^{2} + 11170 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1+3113T+140329926T2+3113p5T3+p10T4 1 + 3113 T + 140329926 T^{2} + 3113 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+7842T+247022914T2+7842p5T3+p10T4 1 + 7842 T + 247022914 T^{2} + 7842 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+12629T+333109116T2+12629p5T3+p10T4 1 + 12629 T + 333109116 T^{2} + 12629 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 19576T+479320714T29576p5T3+p10T4 1 - 9576 T + 479320714 T^{2} - 9576 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 113395T+786785182T213395p5T3+p10T4 1 - 13395 T + 786785182 T^{2} - 13395 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+47521T+1225624018T2+47521p5T3+p10T4 1 + 47521 T + 1225624018 T^{2} + 47521 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+63652T+2359366478T2+63652p5T3+p10T4 1 + 63652 T + 2359366478 T^{2} + 63652 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 144541T+3121830628T244541p5T3+p10T4 1 - 44541 T + 3121830628 T^{2} - 44541 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+125840T+7124822602T2+125840p5T3+p10T4 1 + 125840 T + 7124822602 T^{2} + 125840 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 16039T+3156837796T26039p5T3+p10T4 1 - 6039 T + 3156837796 T^{2} - 6039 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 117588T+1504957625T217588p5T3+p10T4 1 - 17588 T + 1504957625 T^{2} - 17588 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 139325T+1663856032T239325p5T3+p10T4 1 - 39325 T + 1663856032 T^{2} - 39325 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 183082T+12893200018T283082p5T3+p10T4 1 - 83082 T + 12893200018 T^{2} - 83082 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 11905pT+241552312pT21905p6T3+p10T4 1 - 1905 p T + 241552312 p T^{2} - 1905 p^{6} T^{3} + p^{10} T^{4}
show more
show less
   L(s)=p j=14(1αj,pps)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

−9.402108979184850373613484773864, −9.322367494442466004809934437308, −8.689590071441666485042366219094, −8.671521369871877751444029150487, −7.898167524291616087080901357767, −7.61964196192205606873634900900, −7.07432798677968991427662973035, −6.55529239654547634993881089896, −6.00914371027962963478502220060, −5.74868190524297496600881490007, −4.78133924744764873514058081633, −4.73019311604486794483657969696, −3.70060304537933683646842776416, −3.63558930581180187225155524084, −2.70351092046861496668028830351, −2.31172643502052337788849761468, −1.78400884788701691630491324446, −1.44267253025453914559143419610, 0, 0, 1.44267253025453914559143419610, 1.78400884788701691630491324446, 2.31172643502052337788849761468, 2.70351092046861496668028830351, 3.63558930581180187225155524084, 3.70060304537933683646842776416, 4.73019311604486794483657969696, 4.78133924744764873514058081633, 5.74868190524297496600881490007, 6.00914371027962963478502220060, 6.55529239654547634993881089896, 7.07432798677968991427662973035, 7.61964196192205606873634900900, 7.898167524291616087080901357767, 8.671521369871877751444029150487, 8.689590071441666485042366219094, 9.322367494442466004809934437308, 9.402108979184850373613484773864

Graph of the ZZ-function along the critical line