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Properties

Label 4-60e3-1.1-c1e2-0-7
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 + 4·19-s − 20-s + 8·23-s − 3·24-s + 25-s + 27-s + 30-s + 5·32-s − 36-s + 4·38-s − 3·40-s − 20·43-s + 45-s + 8·46-s + 8·47-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 + 0.917·19-s − 0.223·20-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.182·30-s + 0.883·32-s − 1/6·36-s + 0.648·38-s − 0.474·40-s − 3.04·43-s + 0.149·45-s + 1.17·46-s + 1.16·47-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 2.7620784932.762078493
L(12)L(\frac12) ≈\approx 2.7620784932.762078493
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 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
37C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
41C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+8T+pT2)(1+12T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
53C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1−62T2+p2T4 1 - 62 T^{2} + p^{2} T^{4}
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−8T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−10T+pT2)(1−6T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} )
79C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
83C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
89C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 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

−9.082436148341928727121121826518, −8.591346208108976873907943329677, −8.225191928409399460628428045239, −7.64953063828459655550390295412, −6.98398015254135592858985532646, −6.67648428988813377217822323982, −6.05640702866163324913475794325, −5.40296211543658029401315427475, −4.98493646668412576806366292284, −4.71016738217268327245081439923, −3.76684592157047111225216012451, −3.38441085465503245342385480290, −2.86936537122710371154524101951, −2.03884879549215296954331062818, −0.959503861242509782436814994634, 0.959503861242509782436814994634, 2.03884879549215296954331062818, 2.86936537122710371154524101951, 3.38441085465503245342385480290, 3.76684592157047111225216012451, 4.71016738217268327245081439923, 4.98493646668412576806366292284, 5.40296211543658029401315427475, 6.05640702866163324913475794325, 6.67648428988813377217822323982, 6.98398015254135592858985532646, 7.64953063828459655550390295412, 8.225191928409399460628428045239, 8.591346208108976873907943329677, 9.082436148341928727121121826518

Graph of the ZZ-function along the critical line