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Properties

Label 4-60e3-1.1-c1e2-0-21
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 + 9-s − 2·12-s − 2·13-s − 15-s + 4·16-s + 2·20-s + 25-s + 27-s − 8·31-s − 2·36-s + 10·37-s − 2·39-s − 18·41-s + 4·43-s − 45-s + 4·48-s + 2·49-s + 4·52-s − 12·53-s + 2·60-s − 8·64-s + 2·65-s − 8·67-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s + 0.192·27-s − 1.43·31-s − 1/3·36-s + 1.64·37-s − 0.320·39-s − 2.81·41-s + 0.609·43-s − 0.149·45-s + 0.577·48-s + 2/7·49-s + 0.554·52-s − 1.64·53-s + 0.258·60-s − 64-s + 0.248·65-s − 0.977·67-s + 0.115·75-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}
3C1C_1 1−T 1 - T
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 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−14T2+p2T4 1 - 14 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+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1−8T+pT2)(1−2T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1+6T+pT2)(1+12T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
97C22C_2^2 1−170T2+p2T4 1 - 170 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

−8.639234930650642765620751129680, −8.523775257728889674307296197910, −7.895646387548021642457581481954, −7.43218262021209478220184602720, −7.17125924843781199120084644241, −6.32168167241113410404567831150, −5.88169747075832064862554687099, −5.10907947613849939441179482998, −4.77290465581652273268316475331, −4.24331545329566705598520649279, −3.54614090615107725449531004688, −3.21991610284210338990303496611, −2.30332128407427443161963864401, −1.36328580534491248673091037460, 0, 1.36328580534491248673091037460, 2.30332128407427443161963864401, 3.21991610284210338990303496611, 3.54614090615107725449531004688, 4.24331545329566705598520649279, 4.77290465581652273268316475331, 5.10907947613849939441179482998, 5.88169747075832064862554687099, 6.32168167241113410404567831150, 7.17125924843781199120084644241, 7.43218262021209478220184602720, 7.895646387548021642457581481954, 8.523775257728889674307296197910, 8.639234930650642765620751129680

Graph of the ZZ-function along the critical line