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Properties

Label 4-60e3-1.1-c1e2-0-29
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  + 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 + 25-s − 27-s + 2·30-s − 20·31-s − 8·32-s + 2·36-s − 45-s + 4·48-s − 10·49-s + 2·50-s + 20·53-s − 2·54-s + 2·60-s − 40·62-s − 8·64-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 + 1/5·25-s − 0.192·27-s + 0.365·30-s − 3.59·31-s − 1.41·32-s + 1/3·36-s − 0.149·45-s + 0.577·48-s − 1.42·49-s + 0.282·50-s + 2.74·53-s − 0.272·54-s + 0.258·60-s − 5.08·62-s − 64-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−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} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1−30T2+p2T4 1 - 30 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 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
37C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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

−8.796347544631119103074783647720, −8.385333539337096037266712335397, −7.47453418205941725695143179171, −7.27017097594245566397215443793, −6.84957657199386693405152566680, −6.16619991592203718868276685129, −5.68018452289742730764555051785, −5.34999487270269556130717190838, −4.87764075487564301030729914879, −4.11875786665161050620939976019, −3.85363731820904269766144976313, −3.26464349646293471940721383930, −2.45982269552428865640106975011, −1.61170310790947852080742475943, 0, 1.61170310790947852080742475943, 2.45982269552428865640106975011, 3.26464349646293471940721383930, 3.85363731820904269766144976313, 4.11875786665161050620939976019, 4.87764075487564301030729914879, 5.34999487270269556130717190838, 5.68018452289742730764555051785, 6.16619991592203718868276685129, 6.84957657199386693405152566680, 7.27017097594245566397215443793, 7.47453418205941725695143179171, 8.385333539337096037266712335397, 8.796347544631119103074783647720

Graph of the ZZ-function along the critical line