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Properties

Label 4-60e3-1.1-c1e2-0-24
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-s + 2·3-s − 4-s − 5-s − 2·6-s + 3·8-s + 9-s + 10-s − 2·12-s + 3·13-s − 2·15-s − 16-s − 18-s + 20-s + 6·24-s + 25-s − 3·26-s − 4·27-s + 2·30-s − 5·31-s − 5·32-s − 36-s − 21·37-s + 6·39-s − 3·40-s − 12·41-s − 6·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.832·13-s − 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.223·20-s + 1.22·24-s + 1/5·25-s − 0.588·26-s − 0.769·27-s + 0.365·30-s − 0.898·31-s − 0.883·32-s − 1/6·36-s − 3.45·37-s + 0.960·39-s − 0.474·40-s − 1.87·41-s − 0.914·43-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+T+pT2 1 + T + p T^{2}
3C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
5C1C_1 1+T 1 + T
good7C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
29C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+T+pT2)(1+4T+pT2) ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+10T+pT2)(1+11T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} )
41C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2×\timesC2C_2 (1+T+pT2)(1+5T+pT2) ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} )
47C22C_2^2 1−48T2+p2T4 1 - 48 T^{2} + p^{2} T^{4}
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+66T2+p2T4 1 + 66 T^{2} + p^{2} T^{4}
61C22C_2^2 1+29T2+p2T4 1 + 29 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+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )
73C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
83C2C_2×\timesC2C_2 (1−6T+pT2)(1+9T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+8T+pT2)(1+13T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} )
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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

−8.699826047687082336822058020289, −8.407925194342252611576426328430, −8.231152763340728022945344952135, −7.47870553299200018880941726522, −7.10715731494937986420727448761, −6.70285091451097027770080096620, −5.83393016259382569689004866228, −5.17580431414177339059990269329, −4.86833177044701056643889722537, −3.81474994039728055277844345254, −3.68885867024609674771432523082, −3.15262185397536829081504012922, −2.04035771938488058529775286099, −1.49891342858877232756243501199, 0, 1.49891342858877232756243501199, 2.04035771938488058529775286099, 3.15262185397536829081504012922, 3.68885867024609674771432523082, 3.81474994039728055277844345254, 4.86833177044701056643889722537, 5.17580431414177339059990269329, 5.83393016259382569689004866228, 6.70285091451097027770080096620, 7.10715731494937986420727448761, 7.47870553299200018880941726522, 8.231152763340728022945344952135, 8.407925194342252611576426328430, 8.699826047687082336822058020289

Graph of the ZZ-function along the critical line