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Properties

Label 4-60e3-1.1-c1e2-0-12
Degree 44
Conductor 216000216000
Sign 11
Analytic cond. 13.772313.7723
Root an. cond. 1.926421.92642
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 9-s − 2·10-s + 2·12-s − 15-s − 4·16-s + 2·18-s − 2·20-s + 12·23-s + 25-s + 27-s + 10·29-s − 2·30-s − 8·32-s + 2·36-s + 8·43-s − 45-s + 24·46-s + 4·47-s − 4·48-s − 10·49-s + 2·50-s − 8·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s − 0.258·15-s − 16-s + 0.471·18-s − 0.447·20-s + 2.50·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.365·30-s − 1.41·32-s + 1/3·36-s + 1.21·43-s − 0.149·45-s + 3.53·46-s + 0.583·47-s − 0.577·48-s − 1.42·49-s + 0.282·50-s − 1.09·53-s + ⋯

Functional equation

Λ(s)=(216000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(216000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 216000216000    =    26⋅33⋅532^{6} \cdot 3^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 13.772313.7723
Root analytic conductor: 1.926421.92642
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 216000, ( :1/2,1/2), 1)(4,\ 216000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1620690314.162069031
L(12)L(\frac12) ≈\approx 4.1620690314.162069031
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)
bad2C2C_2 1−pT+pT2 1 - p T + p T^{2}
3C1C_1 1−T 1 - T
5C1C_1 1+T 1 + T
good7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C2C_2×\timesC2C_2 (1−10T+pT2)(1+pT2) ( 1 - 10 T + p T^{2} )( 1 + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
41C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2×\timesC2C_2 (1−12T+pT2)(1+8T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−6T+pT2)(1+14T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C2C_2×\timesC2C_2 (1−14T+pT2)(1−4T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} )
79C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
83C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
89C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{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

−8.963134976875332410550837868755, −8.657968102763697321620128690150, −7.998362829294470925327343124130, −7.53885251381440898433031798219, −7.01325793013184944469112654608, −6.50757219156439507333081149228, −6.21841312997577567139299736727, −5.35930529872856387675389024109, −4.87191825277757691038211661892, −4.61465994793805005819745900336, −3.96187051825471841097070685011, −3.20581269290493402875849813307, −3.01888100048798853558195810431, −2.28832826282711051669350915460, −1.06272008942114379011094572988, 1.06272008942114379011094572988, 2.28832826282711051669350915460, 3.01888100048798853558195810431, 3.20581269290493402875849813307, 3.96187051825471841097070685011, 4.61465994793805005819745900336, 4.87191825277757691038211661892, 5.35930529872856387675389024109, 6.21841312997577567139299736727, 6.50757219156439507333081149228, 7.01325793013184944469112654608, 7.53885251381440898433031798219, 7.998362829294470925327343124130, 8.657968102763697321620128690150, 8.963134976875332410550837868755

Graph of the ZZ-function along the critical line