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Properties

Label 4-49392-1.1-c1e2-0-5
Degree 44
Conductor 4939249392
Sign −1-1
Analytic cond. 3.149273.14927
Root an. cond. 1.332141.33214
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 3·9-s + 2·12-s − 14-s − 16-s − 3·18-s − 8·19-s − 2·21-s − 6·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s − 5·32-s − 3·36-s + 12·37-s + 8·38-s + 2·42-s + 2·48-s + 49-s + 6·50-s + 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s + 0.577·12-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 1.83·19-s − 0.436·21-s − 1.22·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s − 0.883·32-s − 1/2·36-s + 1.97·37-s + 1.29·38-s + 0.308·42-s + 0.288·48-s + 1/7·49-s + 0.848·50-s + 1.64·53-s + ⋯

Functional equation

Λ(s)=(49392s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(49392s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4939249392    =    24⋅32⋅732^{4} \cdot 3^{2} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 3.149273.14927
Root analytic conductor: 1.332141.33214
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 49392, ( :1/2,1/2), −1)(4,\ 49392,\ (\ :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}
3C1C_1 (1+T)2 ( 1 + T )^{2}
7C1C_1 1−T 1 - T
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−9.863214598051417092763914791206, −9.198895146278336601365818133221, −9.163045893798789171919999202805, −8.068098981215555715874277637796, −7.977686193886306753143677629185, −7.38325958034671995701748313853, −6.53484921474790582970192984243, −6.19643457342138342025418009798, −5.48510765590071063887274763802, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −4.01671863219499060051377207353, −2.39234840548789475685588947322, −1.40786771277750203137263322761, 0, 1.40786771277750203137263322761, 2.39234840548789475685588947322, 4.01671863219499060051377207353, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.48510765590071063887274763802, 6.19643457342138342025418009798, 6.53484921474790582970192984243, 7.38325958034671995701748313853, 7.977686193886306753143677629185, 8.068098981215555715874277637796, 9.163045893798789171919999202805, 9.198895146278336601365818133221, 9.863214598051417092763914791206

Graph of the ZZ-function along the critical line