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Properties

Label 4-60e3-1.1-c1e2-0-4
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 − 6·13-s + 15-s − 16-s + 18-s − 20-s − 3·24-s + 25-s − 6·26-s + 27-s + 30-s + 16·31-s + 5·32-s − 36-s + 6·37-s − 6·39-s − 3·40-s + 6·41-s + 45-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 − 1.66·13-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.182·30-s + 2.87·31-s + 0.883·32-s − 1/6·36-s + 0.986·37-s − 0.960·39-s − 0.474·40-s + 0.937·41-s + 0.149·45-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 2.4383545772.438354577
L(12)L(\frac12) ≈\approx 2.4383545772.438354577
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}
13C2C_2×\timesC2C_2 (1+pT2)(1+6T+pT2) ( 1 + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1−4T+pT2)(1−2T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−4T+pT2)(1−2T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )
43C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
59C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
61C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+pT2)(1+12T+pT2) ( 1 + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
73C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
89C2C_2×\timesC2C_2 (1−14T+pT2)(1+8T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C22C_2^2 1−74T2+p2T4 1 - 74 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

−9.136829845041719869803204909610, −8.572448663066506355962139996664, −8.091898104800528160801263424980, −7.70383442746954505009106361716, −7.12485387632655359718601026080, −6.48835408883971268939119199881, −6.13431711419477769810541152571, −5.48960454448931742334640145872, −4.90867080535439970632935053629, −4.52844504199261714499832165926, −4.12229626934867515006504594626, −3.22087665149485841437962575171, −2.68492287371980531455329544918, −2.26607927320469213703160857526, −0.864946238466998714244148432182, 0.864946238466998714244148432182, 2.26607927320469213703160857526, 2.68492287371980531455329544918, 3.22087665149485841437962575171, 4.12229626934867515006504594626, 4.52844504199261714499832165926, 4.90867080535439970632935053629, 5.48960454448931742334640145872, 6.13431711419477769810541152571, 6.48835408883971268939119199881, 7.12485387632655359718601026080, 7.70383442746954505009106361716, 8.091898104800528160801263424980, 8.572448663066506355962139996664, 9.136829845041719869803204909610

Graph of the ZZ-function along the critical line