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Properties

Label 4-60e3-1.1-c1e2-0-23
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 (1−2T+pT2)(1+T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 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 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 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+T+pT2)(1+10T+pT2) ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} )
47C22C_2^2 1−23T2+p2T4 1 - 23 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−6T+pT2)(1+3T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 3 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+4T+pT2)(1+13T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 13 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 (1−12T+pT2)(1−6T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 6 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

−9.029973055593751794701534273253, −8.437580779945766317139841544546, −8.010436785697642736041423600877, −7.42819112305891538658843940694, −7.05628584860540280837272529827, −6.24765594036739310828616004723, −5.64808565673135417116444211324, −5.51206865917945044451462215346, −4.79057202611980715265646061167, −4.18535974972689387750755568412, −3.47343971131727371766406808196, −3.21541107080296658914621949913, −2.23362429346088432123663168401, −1.48731869657594409478328408543, 0, 1.48731869657594409478328408543, 2.23362429346088432123663168401, 3.21541107080296658914621949913, 3.47343971131727371766406808196, 4.18535974972689387750755568412, 4.79057202611980715265646061167, 5.51206865917945044451462215346, 5.64808565673135417116444211324, 6.24765594036739310828616004723, 7.05628584860540280837272529827, 7.42819112305891538658843940694, 8.010436785697642736041423600877, 8.437580779945766317139841544546, 9.029973055593751794701534273253

Graph of the ZZ-function along the critical line