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Properties

Label 4-60e3-1.1-c1e2-0-2
Degree 44
Conductor 216000216000
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 12-s − 15-s − 16-s − 18-s − 6·19-s + 20-s + 12·23-s + 3·24-s + 25-s + 27-s + 10·29-s + 30-s − 5·32-s − 36-s + 6·38-s − 3·40-s − 10·43-s − 45-s − 12·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 2.50·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.883·32-s − 1/6·36-s + 0.973·38-s − 0.474·40-s − 1.52·43-s − 0.149·45-s − 1.76·46-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: 11
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: 00
Selberg data: (4, 216000, ( :1/2,1/2), 1)(4,\ 216000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0968800351.096880035
L(12)L(\frac12) ≈\approx 1.0968800351.096880035
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+T+pT2 1 + T + 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+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
13C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
17C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1+pT2)(1+6T+pT2) ( 1 + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C2C_2×\timesC2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C22C_2^2 1+24T2+p2T4 1 + 24 T^{2} + p^{2} T^{4}
41C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1−32T2+p2T4 1 - 32 T^{2} + p^{2} T^{4}
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−12T+pT2)(1+6T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−8T+pT2)(1−2T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−16T+pT2)(1+10T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
83C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
89C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+10T+pT2)(1+14T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 14 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

−8.892399351428167215131965300837, −8.544030733817935620568279576532, −8.265344842528523210883241623812, −7.902520218753514335027686044494, −7.15209964462953520184720736322, −6.72521606672367475647233337637, −6.53468878473785351919474017241, −5.38535674461546969487184019211, −4.97535280951748912982547092092, −4.50946846798418509258280508548, −3.95532995302561227890190721683, −3.23904594666351099422428199201, −2.66889019243147502869235143114, −1.67986934782382229520815967235, −0.75507799196260966330798213842, 0.75507799196260966330798213842, 1.67986934782382229520815967235, 2.66889019243147502869235143114, 3.23904594666351099422428199201, 3.95532995302561227890190721683, 4.50946846798418509258280508548, 4.97535280951748912982547092092, 5.38535674461546969487184019211, 6.53468878473785351919474017241, 6.72521606672367475647233337637, 7.15209964462953520184720736322, 7.902520218753514335027686044494, 8.265344842528523210883241623812, 8.544030733817935620568279576532, 8.892399351428167215131965300837

Graph of the ZZ-function along the critical line