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Properties

Label 4-294e2-1.1-c5e2-0-21
Degree 44
Conductor 8643686436
Sign 11
Analytic cond. 2223.392223.39
Root an. cond. 6.866796.86679
Motivic weight 55
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  + 8·2-s + 18·3-s + 48·4-s + 18·5-s + 144·6-s + 256·8-s + 243·9-s + 144·10-s + 2·11-s + 864·12-s + 288·13-s + 324·15-s + 1.28e3·16-s + 1.53e3·17-s + 1.94e3·18-s + 1.18e3·19-s + 864·20-s + 16·22-s + 3.39e3·23-s + 4.60e3·24-s − 1.30e3·25-s + 2.30e3·26-s + 2.91e3·27-s − 3.97e3·29-s + 2.59e3·30-s + 7.59e3·31-s + 6.14e3·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.321·5-s + 1.63·6-s + 1.41·8-s + 9-s + 0.455·10-s + 0.00498·11-s + 1.73·12-s + 0.472·13-s + 0.371·15-s + 5/4·16-s + 1.28·17-s + 1.41·18-s + 0.754·19-s + 0.482·20-s + 0.00704·22-s + 1.33·23-s + 1.63·24-s − 0.416·25-s + 0.668·26-s + 0.769·27-s − 0.877·29-s + 0.525·30-s + 1.41·31-s + 1.06·32-s + ⋯

Functional equation

Λ(s)=(86436s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(86436s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8643686436    =    22⋅32⋅742^{2} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 2223.392223.39
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 86436, ( :5/2,5/2), 1)(4,\ 86436,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 19.6280461319.62804613
L(12)L(\frac12) ≈\approx 19.6280461319.62804613
L(72)L(\frac{7}{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)
bad2C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
3C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
7 1 1
good5D4D_{4} 1−18T+1626T2−18p5T3+p10T4 1 - 18 T + 1626 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1−2T−59002T2−2p5T3+p10T4 1 - 2 T - 59002 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1−288T+688042T2−288p5T3+p10T4 1 - 288 T + 688042 T^{2} - 288 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−90pT+2366314T2−90p6T3+p10T4 1 - 90 p T + 2366314 T^{2} - 90 p^{6} T^{3} + p^{10} T^{4}
19D4D_{4} 1−1188T+4834534T2−1188p5T3+p10T4 1 - 1188 T + 4834534 T^{2} - 1188 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1−3390T+6218086T2−3390p5T3+p10T4 1 - 3390 T + 6218086 T^{2} - 3390 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+3976T+31254662T2+3976p5T3+p10T4 1 + 3976 T + 31254662 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1−7596T+61727326T2−7596p5T3+p10T4 1 - 7596 T + 61727326 T^{2} - 7596 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−2688T+126774470T2−2688p5T3+p10T4 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1−36630T+559995322T2−36630p5T3+p10T4 1 - 36630 T + 559995322 T^{2} - 36630 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+23032T+329072262T2+23032p5T3+p10T4 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−864T+46643358T2−864p5T3+p10T4 1 - 864 T + 46643358 T^{2} - 864 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+32920T+983844566T2+32920p5T3+p10T4 1 + 32920 T + 983844566 T^{2} + 32920 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+26712T+697343334T2+26712p5T3+p10T4 1 + 26712 T + 697343334 T^{2} + 26712 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1−20412T+1230091258T2−20412p5T3+p10T4 1 - 20412 T + 1230091258 T^{2} - 20412 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+36172T+33148290pT2+36172p5T3+p10T4 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−73706T+3356433686T2−73706p5T3+p10T4 1 - 73706 T + 3356433686 T^{2} - 73706 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+74772T+3993671602T2+74772p5T3+p10T4 1 + 74772 T + 3993671602 T^{2} + 74772 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+23116T+6249589662T2+23116p5T3+p10T4 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1−147816T+13316083030T2−147816p5T3+p10T4 1 - 147816 T + 13316083030 T^{2} - 147816 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1−164646T+17878567722T2−164646p5T3+p10T4 1 - 164646 T + 17878567722 T^{2} - 164646 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1−162036T+20528488258T2−162036p5T3+p10T4 1 - 162036 T + 20528488258 T^{2} - 162036 p^{5} T^{3} + p^{10} 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

−11.03126449600863208109055982126, −11.00471394897125006080699569534, −10.21015340067008375016439628505, −9.724665766568722305175836917371, −9.356517174017467275586622541514, −8.849245838852459243027358653511, −8.030569639097068673788683219051, −7.73122648234537336719519786689, −7.38916085927152052026579388151, −6.65197399475395838663448612034, −6.08625616193183677395384595949, −5.73821327159221643919228818660, −4.79343678164705587341720345002, −4.74491002185730943442141796740, −3.63689787001847966387176938758, −3.49898359280308871950861336258, −2.80945490551206674681251538260, −2.32358838506610979906653471636, −1.40505268813211093398721159124, −0.950314725667538151007569784201, 0.950314725667538151007569784201, 1.40505268813211093398721159124, 2.32358838506610979906653471636, 2.80945490551206674681251538260, 3.49898359280308871950861336258, 3.63689787001847966387176938758, 4.74491002185730943442141796740, 4.79343678164705587341720345002, 5.73821327159221643919228818660, 6.08625616193183677395384595949, 6.65197399475395838663448612034, 7.38916085927152052026579388151, 7.73122648234537336719519786689, 8.030569639097068673788683219051, 8.849245838852459243027358653511, 9.356517174017467275586622541514, 9.724665766568722305175836917371, 10.21015340067008375016439628505, 11.00471394897125006080699569534, 11.03126449600863208109055982126

Graph of the ZZ-function along the critical line