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Properties

Label 4-60e3-1.1-c1e2-0-11
Degree 44
Conductor 216000216000
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s − 2·9-s + 2·12-s − 13-s + 15-s + 4·16-s + 2·20-s + 25-s + 5·27-s − 14·31-s + 4·36-s + 8·37-s + 39-s + 9·41-s + 11·43-s + 2·45-s − 4·48-s + 2·49-s + 2·52-s − 3·53-s − 2·60-s − 8·64-s + 65-s + 17·67-s − 9·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s + 0.962·27-s − 2.51·31-s + 2/3·36-s + 1.31·37-s + 0.160·39-s + 1.40·41-s + 1.67·43-s + 0.298·45-s − 0.577·48-s + 2/7·49-s + 0.277·52-s − 0.412·53-s − 0.258·60-s − 64-s + 0.124·65-s + 2.07·67-s − 1.06·71-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: −1-1
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: 11
Selberg data: (4, 216000, ( :1/2,1/2), −1)(4,\ 216000,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+pT2 1 + p T^{2}
3C2C_2 1+T+pT2 1 + T + p T^{2}
5C1C_1 1+T 1 + T
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−T+pT2)(1+2T+pT2) ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
19C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
23C22C_2^2 1−29T2+p2T4 1 - 29 T^{2} + p^{2} T^{4}
29C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1−3T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
47C22C_2^2 1−23T2+p2T4 1 - 23 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C22C_2^2 1+31T2+p2T4 1 + 31 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−13T+pT2)(1−4T+pT2) ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−6T+pT2)(1+15T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} )
73C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2×\timesC2C_2 (1+6T+pT2)(1+12T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+3T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} )
97C22C_2^2 1+136T2+p2T4 1 + 136 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

−8.933704118300936802591523900027, −8.432953852457324344679654943509, −7.76492047294845852924304826356, −7.50792686685256150029334773001, −6.99299729246842230861448013078, −6.16613662934134908604003916356, −5.71806650566000179487289720171, −5.48323312196914960455202802870, −4.76738470375839414337236762761, −4.25906761632603670109845527152, −3.79996533598904598436090270715, −3.06483335293129371128757025033, −2.33517872487363433910125537907, −1.04459128719907683856615859851, 0, 1.04459128719907683856615859851, 2.33517872487363433910125537907, 3.06483335293129371128757025033, 3.79996533598904598436090270715, 4.25906761632603670109845527152, 4.76738470375839414337236762761, 5.48323312196914960455202802870, 5.71806650566000179487289720171, 6.16613662934134908604003916356, 6.99299729246842230861448013078, 7.50792686685256150029334773001, 7.76492047294845852924304826356, 8.432953852457324344679654943509, 8.933704118300936802591523900027

Graph of the ZZ-function along the critical line