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Properties

Label 4-60e3-1.1-c1e2-0-0
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-s − 3-s − 4-s − 5-s + 6-s + 3·8-s − 2·9-s + 10-s + 12-s + 15-s − 16-s + 2·18-s + 20-s − 3·24-s + 25-s + 5·27-s − 30-s + 7·31-s − 5·32-s + 2·36-s − 9·37-s − 3·40-s + 3·41-s − 6·43-s + 2·45-s + 48-s + 11·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.471·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.962·27-s − 0.182·30-s + 1.25·31-s − 0.883·32-s + 1/3·36-s − 1.47·37-s − 0.474·40-s + 0.468·41-s − 0.914·43-s + 0.298·45-s + 0.144·48-s + 11/7·49-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 0.47453730840.4745373084
L(12)L(\frac12) ≈\approx 0.47453730840.4745373084
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+T+pT2 1 + T + p T^{2}
3C2C_2 1+T+pT2 1 + T + p T^{2}
5C1C_1 1+T 1 + T
good7C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C22C_2^2 1+12T2+p2T4 1 + 12 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−5T+pT2)(1−2T+pT2) ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−5T+pT2)(1+2T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
47C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−10T+pT2)(1+5T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} )
59C22C_2^2 1+15T2+p2T4 1 + 15 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×\timesC2C_2 (1+9T+pT2)(1+15T+pT2) ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−T+pT2)(1+16T+pT2) ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} )
73C22C_2^2 1−140T2+p2T4 1 - 140 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−10T+pT2)(1−4T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−9T+pT2)(1−6T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+T+pT2)(1+14T+pT2) ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} )
97C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} 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

−9.135302092333353652652186528333, −8.568822291342378708563372922224, −8.194417309724579054838462347537, −7.77531817530388024228130269937, −7.11431484901848047337285972916, −6.84899310638551761424185074277, −6.01612086119737147435067066575, −5.69886437390822823949011200716, −4.99207371310814885896269263171, −4.63021633725152443209733084264, −4.01480313584461926637256837089, −3.31874751359050490312370733802, −2.61237471789384733489531182439, −1.53647528102741022128099740579, −0.52408110145612652103271752356, 0.52408110145612652103271752356, 1.53647528102741022128099740579, 2.61237471789384733489531182439, 3.31874751359050490312370733802, 4.01480313584461926637256837089, 4.63021633725152443209733084264, 4.99207371310814885896269263171, 5.69886437390822823949011200716, 6.01612086119737147435067066575, 6.84899310638551761424185074277, 7.11431484901848047337285972916, 7.77531817530388024228130269937, 8.194417309724579054838462347537, 8.568822291342378708563372922224, 9.135302092333353652652186528333

Graph of the ZZ-function along the critical line