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Properties

Label 4-60e3-1.1-c1e2-0-9
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 no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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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 + 4·13-s + 15-s − 16-s − 4·17-s + 18-s + 8·19-s − 20-s − 3·24-s + 25-s + 4·26-s + 27-s − 4·29-s + 30-s + 5·32-s − 4·34-s − 36-s + 20·37-s + 8·38-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 + 1.10·13-s + 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.742·29-s + 0.182·30-s + 0.883·32-s − 0.685·34-s − 1/6·36-s + 3.28·37-s + 1.29·38-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: no
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 2.8652300042.865230004
L(12)L(\frac12) ≈\approx 2.8652300042.865230004
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+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−8.881388845996159681294188149673, −8.843956595114680827452383410006, −8.084180434635331166491599942631, −7.68003844385765985083943135134, −7.16842258862004714094187855827, −6.44491636719189125157029592476, −5.86497653842965752491328535183, −5.83817914073203780391805721828, −4.88120268416850114670329243377, −4.52160444884874669934496061646, −3.99423297820309292802284752251, −3.16799571044027062920603606790, −3.01549784140686353642765162463, −1.98364241717056996622491365555, −0.991628865642481429121344680829, 0.991628865642481429121344680829, 1.98364241717056996622491365555, 3.01549784140686353642765162463, 3.16799571044027062920603606790, 3.99423297820309292802284752251, 4.52160444884874669934496061646, 4.88120268416850114670329243377, 5.83817914073203780391805721828, 5.86497653842965752491328535183, 6.44491636719189125157029592476, 7.16842258862004714094187855827, 7.68003844385765985083943135134, 8.084180434635331166491599942631, 8.843956595114680827452383410006, 8.881388845996159681294188149673

Graph of the ZZ-function along the critical line