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Properties

Label 4-60e3-1.1-c1e2-0-26
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 − 2·9-s − 2·10-s − 2·12-s + 3·13-s + 15-s − 4·16-s − 4·18-s − 2·20-s + 25-s + 6·26-s + 5·27-s + 2·30-s − 2·31-s − 8·32-s − 4·36-s − 12·37-s − 3·39-s − 3·41-s + 3·43-s + 2·45-s + 4·48-s − 10·49-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 2/3·9-s − 0.632·10-s − 0.577·12-s + 0.832·13-s + 0.258·15-s − 16-s − 0.942·18-s − 0.447·20-s + 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.365·30-s − 0.359·31-s − 1.41·32-s − 2/3·36-s − 1.97·37-s − 0.480·39-s − 0.468·41-s + 0.457·43-s + 0.298·45-s + 0.577·48-s − 1.42·49-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: −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}
3C2C_2 1+T+pT2 1 + T + p T^{2}
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} )
11C22C_2^2 1−12T2+p2T4 1 - 12 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−6T+pT2)(1+3T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
23C22C_2^2 1−21T2+p2T4 1 - 21 T^{2} + p^{2} T^{4}
29C22C_2^2 1−32T2+p2T4 1 - 32 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 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 (1−2T+pT2)(1+5T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−2T+pT2)(1−T+pT2) ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} )
47C22C_2^2 1−15T2+p2T4 1 - 15 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−7T+pT2)(1+14T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
61C22C_2^2 1+95T2+p2T4 1 + 95 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−12T+pT2)(1+3T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−5T+pT2)(1+14T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} )
73C22C_2^2 1+100T2+p2T4 1 + 100 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−4T+pT2)(1+14T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 14 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+10T+pT2)(1+11T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} )
97C22C_2^2 1−176T2+p2T4 1 - 176 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

−8.722339061279030502026217557080, −8.326625773390273821916688279470, −7.86104944075813803200111539928, −6.98306956144003059227349427373, −6.80029561603651451983469425798, −6.20925779024294472317126229619, −5.74294810853403922092955208519, −5.32290141168043296240714248598, −4.84739726754712687571159778127, −4.31296215270085071722819888265, −3.57959421488730987783136762088, −3.31251437718625774347934014906, −2.57128120301065922656290658012, −1.55179478107478001093121490033, 0, 1.55179478107478001093121490033, 2.57128120301065922656290658012, 3.31251437718625774347934014906, 3.57959421488730987783136762088, 4.31296215270085071722819888265, 4.84739726754712687571159778127, 5.32290141168043296240714248598, 5.74294810853403922092955208519, 6.20925779024294472317126229619, 6.80029561603651451983469425798, 6.98306956144003059227349427373, 7.86104944075813803200111539928, 8.326625773390273821916688279470, 8.722339061279030502026217557080

Graph of the ZZ-function along the critical line