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Properties

Label 4-160e2-1.1-c3e2-0-4
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 89.119389.1193
Root an. cond. 3.072503.07250
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  + 8·3-s + 10·5-s + 8·7-s + 18·9-s + 64·11-s + 12·13-s + 80·15-s + 4·17-s + 208·19-s + 64·21-s + 120·23-s + 75·25-s − 8·27-s − 292·29-s + 176·31-s + 512·33-s + 80·35-s − 356·37-s + 96·39-s + 100·41-s − 376·43-s + 180·45-s − 280·47-s − 38·49-s + 32·51-s + 316·53-s + 640·55-s + ⋯
L(s)  = 1  + 1.53·3-s + 0.894·5-s + 0.431·7-s + 2/3·9-s + 1.75·11-s + 0.256·13-s + 1.37·15-s + 0.0570·17-s + 2.51·19-s + 0.665·21-s + 1.08·23-s + 3/5·25-s − 0.0570·27-s − 1.86·29-s + 1.01·31-s + 2.70·33-s + 0.386·35-s − 1.58·37-s + 0.394·39-s + 0.380·41-s − 1.33·43-s + 0.596·45-s − 0.868·47-s − 0.110·49-s + 0.0878·51-s + 0.818·53-s + 1.56·55-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 89.119389.1193
Root analytic conductor: 3.072503.07250
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25600, ( :3/2,3/2), 1)(4,\ 25600,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.9257690765.925769076
L(12)L(\frac12) ≈\approx 5.9257690765.925769076
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
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3D4D_{4} 1−8T+46T2−8p3T3+p6T4 1 - 8 T + 46 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−8T+102T2−8p3T3+p6T4 1 - 8 T + 102 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−64T+2822T2−64p3T3+p6T4 1 - 64 T + 2822 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−12T+2894T2−12p3T3+p6T4 1 - 12 T + 2894 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−4T−3994T2−4p3T3+p6T4 1 - 4 T - 3994 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−208T+22998T2−208p3T3+p6T4 1 - 208 T + 22998 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−120T+25030T2−120p3T3+p6T4 1 - 120 T + 25030 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+292T+63950T2+292p3T3+p6T4 1 + 292 T + 63950 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−176T+66462T2−176p3T3+p6T4 1 - 176 T + 66462 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+356T+126846T2+356p3T3+p6T4 1 + 356 T + 126846 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−100T+101942T2−100p3T3+p6T4 1 - 100 T + 101942 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+376T+161502T2+376p3T3+p6T4 1 + 376 T + 161502 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+280T+223190T2+280p3T3+p6T4 1 + 280 T + 223190 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−316T+308894T2−316p3T3+p6T4 1 - 316 T + 308894 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−720T+530758T2−720p3T3+p6T4 1 - 720 T + 530758 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1268T+831342T2+1268p3T3+p6T4 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+744T+686894T2+744p3T3+p6T4 1 + 744 T + 686894 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+48T−424178T2+48p3T3+p6T4 1 + 48 T - 424178 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+940T+653334T2+940p3T3+p6T4 1 + 940 T + 653334 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−32T+276318T2−32p3T3+p6T4 1 - 32 T + 276318 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1592T+1724174T2+1592p3T3+p6T4 1 + 1592 T + 1724174 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+780T+1506742T2+780p3T3+p6T4 1 + 780 T + 1506742 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1220T+2159046T2−1220p3T3+p6T4 1 - 1220 T + 2159046 T^{2} - 1220 p^{3} T^{3} + 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

−12.90057976641195194326403702788, −12.07944666155499279161735114316, −11.59454307411752076025443783844, −11.43786892411146527381082739595, −10.48909510800178765538128060999, −9.946914282481845073682175216502, −9.374549824180792744869599810321, −9.132337644566011188498566344507, −8.724896961806542289458338668585, −8.200018756500394738665247039334, −7.32039364854479990015315683827, −7.14390397592921254683681862828, −6.28480277508495431318534754157, −5.57625104557087285003188415581, −5.01782670845119088910440796014, −4.06655852415350657571465781181, −3.19481007479969433618600058947, −3.02617259156789579275513362149, −1.70856544325521916667768543424, −1.29698493764042189011672068702, 1.29698493764042189011672068702, 1.70856544325521916667768543424, 3.02617259156789579275513362149, 3.19481007479969433618600058947, 4.06655852415350657571465781181, 5.01782670845119088910440796014, 5.57625104557087285003188415581, 6.28480277508495431318534754157, 7.14390397592921254683681862828, 7.32039364854479990015315683827, 8.200018756500394738665247039334, 8.724896961806542289458338668585, 9.132337644566011188498566344507, 9.374549824180792744869599810321, 9.946914282481845073682175216502, 10.48909510800178765538128060999, 11.43786892411146527381082739595, 11.59454307411752076025443783844, 12.07944666155499279161735114316, 12.90057976641195194326403702788

Graph of the ZZ-function along the critical line