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Properties

Label 4-1200e2-1.1-c1e2-0-3
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 91.815691.8156
Root an. cond. 3.095483.09548
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s − 6·11-s + 4·13-s − 12·23-s − 9·27-s + 18·33-s − 16·37-s − 12·39-s + 12·47-s + 14·49-s − 24·59-s + 16·61-s + 36·69-s + 12·71-s − 2·73-s + 9·81-s + 18·83-s − 20·97-s − 36·99-s − 6·107-s − 16·109-s + 48·111-s + 24·117-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 1.80·11-s + 1.10·13-s − 2.50·23-s − 1.73·27-s + 3.13·33-s − 2.63·37-s − 1.92·39-s + 1.75·47-s + 2·49-s − 3.12·59-s + 2.04·61-s + 4.33·69-s + 1.42·71-s − 0.234·73-s + 81-s + 1.97·83-s − 2.03·97-s − 3.61·99-s − 0.580·107-s − 1.53·109-s + 4.55·111-s + 2.21·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 91.815691.8156
Root analytic conductor: 3.095483.09548
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1440000, ( :1/2,1/2), 1)(4,\ 1440000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.37661216000.3766121600
L(12)L(\frac12) ≈\approx 0.37661216000.3766121600
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)
bad2 1 1
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
5 1 1
good7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C22C_2^2 1−55T2+p2T4 1 - 55 T^{2} + p^{2} T^{4}
43C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
47C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
53C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
79C22C_2^2 1−110T2+p2T4 1 - 110 T^{2} + p^{2} T^{4}
83C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
89C22C_2^2 1−151T2+p2T4 1 - 151 T^{2} + p^{2} T^{4}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
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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

−10.24241117420089599362030237578, −9.735622760464094679804352445585, −9.160218818393113271881260638511, −8.764439734700187667739256783256, −8.082374327393994916182205016028, −7.960837988361317902228831190964, −7.48843416880746156683661506366, −6.89091377387839359008903746689, −6.59947152126265751554091188917, −6.04781706637437252987046953342, −5.68392438476744828821678133760, −5.38130014074993360084710856524, −5.10645051458956556813116411809, −4.37457029886652802141706842065, −3.93615750866165581470282315462, −3.56055383961580793173395611144, −2.58519152879164453411555551623, −2.06033238712800569075589944275, −1.29936585137234847375855601110, −0.30969000246082014815069030507, 0.30969000246082014815069030507, 1.29936585137234847375855601110, 2.06033238712800569075589944275, 2.58519152879164453411555551623, 3.56055383961580793173395611144, 3.93615750866165581470282315462, 4.37457029886652802141706842065, 5.10645051458956556813116411809, 5.38130014074993360084710856524, 5.68392438476744828821678133760, 6.04781706637437252987046953342, 6.59947152126265751554091188917, 6.89091377387839359008903746689, 7.48843416880746156683661506366, 7.960837988361317902228831190964, 8.082374327393994916182205016028, 8.764439734700187667739256783256, 9.160218818393113271881260638511, 9.735622760464094679804352445585, 10.24241117420089599362030237578

Graph of the ZZ-function along the critical line