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Properties

Label 4-132300-1.1-c1e2-0-19
Degree 44
Conductor 132300132300
Sign 11
Analytic cond. 8.435568.43556
Root an. cond. 1.704231.70423
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  + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s + 16·19-s + 2·21-s + 25-s + 27-s + 2·28-s − 8·31-s + 36-s − 20·37-s + 4·39-s − 8·43-s + 48-s + 3·49-s + 4·52-s + 16·57-s − 20·61-s + 2·63-s + 64-s − 8·67-s − 20·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 3.67·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 3.28·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s + 2.11·57-s − 2.56·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 2.34·73-s + 0.115·75-s + ⋯

Functional equation

Λ(s)=(132300s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(132300s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 132300132300    =    22⋅33⋅52⋅722^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 8.435568.43556
Root analytic conductor: 1.704231.70423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 132300, ( :1/2,1/2), 1)(4,\ 132300,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7470072172.747007217
L(12)L(\frac12) ≈\approx 2.7470072172.747007217
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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C1C_1 1−T 1 - T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
7C1C_1 (1−T)2 ( 1 - T )^{2}
good11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{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

−9.262397060563074490443203035163, −8.790922026040486061843055845413, −8.581528352643131576230344729464, −7.67845508591579519153392227466, −7.48230039465939347417283743898, −7.19801386358340821477645347390, −6.43968643454495280602009146366, −5.76093596813527562032392850894, −5.20272031164140682979668587117, −4.97103303999261607614028493172, −3.88273199773070623460496754838, −3.22977772739188491587716942834, −3.13125557587754062819456448397, −1.69594451483764738863265937021, −1.41006155845401754086894809680, 1.41006155845401754086894809680, 1.69594451483764738863265937021, 3.13125557587754062819456448397, 3.22977772739188491587716942834, 3.88273199773070623460496754838, 4.97103303999261607614028493172, 5.20272031164140682979668587117, 5.76093596813527562032392850894, 6.43968643454495280602009146366, 7.19801386358340821477645347390, 7.48230039465939347417283743898, 7.67845508591579519153392227466, 8.581528352643131576230344729464, 8.790922026040486061843055845413, 9.262397060563074490443203035163

Graph of the ZZ-function along the critical line