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Properties

Label 4-1200e2-1.1-c1e2-0-20
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 3.6907991683.690799168
L(12)L(\frac12) ≈\approx 3.6907991683.690799168
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

−9.949071215160877709405285288689, −9.388712944607297374453685683482, −9.148258902572960841846009259247, −8.807714535280942665412841301694, −8.255774497202089660090663603470, −7.77666664841938190662447383816, −7.71661740425551270066394270788, −7.34009431470494632789417767851, −6.77560219337028756121239169548, −6.40687562695412033412906492299, −5.54815261604095958672505092947, −5.18343743253181781866862967579, −4.76447631261619758628405568919, −4.34226208320684842695419885883, −3.71331411593505908329064024068, −2.88253801903961845956274903295, −2.85753585158968917578674971710, −2.48302961494227328366943154877, −1.70698974645208637812633518352, −0.73878341732076462904245968158, 0.73878341732076462904245968158, 1.70698974645208637812633518352, 2.48302961494227328366943154877, 2.85753585158968917578674971710, 2.88253801903961845956274903295, 3.71331411593505908329064024068, 4.34226208320684842695419885883, 4.76447631261619758628405568919, 5.18343743253181781866862967579, 5.54815261604095958672505092947, 6.40687562695412033412906492299, 6.77560219337028756121239169548, 7.34009431470494632789417767851, 7.71661740425551270066394270788, 7.77666664841938190662447383816, 8.255774497202089660090663603470, 8.807714535280942665412841301694, 9.148258902572960841846009259247, 9.388712944607297374453685683482, 9.949071215160877709405285288689

Graph of the ZZ-function along the critical line