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Properties

Label 4-540800-1.1-c1e2-0-48
Degree 44
Conductor 540800540800
Sign 11
Analytic cond. 34.481834.4818
Root an. cond. 2.423242.42324
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 + 4-s + 2·5-s + 8-s + 5·9-s + 2·10-s + 5·11-s + 4·13-s + 16-s − 5·17-s + 5·18-s + 2·20-s + 5·22-s − 25-s + 4·26-s + 32-s − 5·34-s + 5·36-s − 16·37-s + 2·40-s + 5·44-s + 10·45-s + 5·49-s − 50-s + 4·52-s + 10·55-s + 10·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 5/3·9-s + 0.632·10-s + 1.50·11-s + 1.10·13-s + 1/4·16-s − 1.21·17-s + 1.17·18-s + 0.447·20-s + 1.06·22-s − 1/5·25-s + 0.784·26-s + 0.176·32-s − 0.857·34-s + 5/6·36-s − 2.63·37-s + 0.316·40-s + 0.753·44-s + 1.49·45-s + 5/7·49-s − 0.141·50-s + 0.554·52-s + 1.34·55-s + 1.30·59-s + ⋯

Functional equation

Λ(s)=(540800s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(540800s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 540800540800    =    27⋅52⋅1322^{7} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 34.481834.4818
Root analytic conductor: 2.423242.42324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 540800, ( :1/2,1/2), 1)(4,\ 540800,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9273542874.927354287
L(12)L(\frac12) ≈\approx 4.9273542874.927354287
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 1−T 1 - T
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
13C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
7C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−5T+pT2)(1+pT2) ( 1 - 5 T + p T^{2} )( 1 + p T^{2} )
17C2C_2×\timesC2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
19C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
31C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1+75T2+p2T4 1 + 75 T^{2} + p^{2} T^{4}
53C22C_2^2 1+35T2+p2T4 1 + 35 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−9T+pT2)(1−T+pT2) ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} )
61C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+7T+pT2)(1+12T+pT2) ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
73C22C_2^2 1+105T2+p2T4 1 + 105 T^{2} + p^{2} T^{4}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−T+pT2)(1+4T+pT2) ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} )
89C22C_2^2 1−127T2+p2T4 1 - 127 T^{2} + p^{2} T^{4}
97C22C_2^2 1−50T2+p2T4 1 - 50 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

−8.693106897828908108247753364178, −7.952030872113872173367866691847, −7.24400594316398621315503187668, −7.01866428971125593812595099427, −6.58741997017797074790016737973, −6.22964068496149410178266735509, −5.78742587874235716738152627165, −5.16293979433082427180753488762, −4.60177403178348512577473878035, −4.10660632241969598906900634621, −3.79277034805909303170184144047, −3.22426006217808700760784232060, −2.14045533678656230614910911624, −1.74171000561321275859235095117, −1.20068436925435355317228040963, 1.20068436925435355317228040963, 1.74171000561321275859235095117, 2.14045533678656230614910911624, 3.22426006217808700760784232060, 3.79277034805909303170184144047, 4.10660632241969598906900634621, 4.60177403178348512577473878035, 5.16293979433082427180753488762, 5.78742587874235716738152627165, 6.22964068496149410178266735509, 6.58741997017797074790016737973, 7.01866428971125593812595099427, 7.24400594316398621315503187668, 7.952030872113872173367866691847, 8.693106897828908108247753364178

Graph of the ZZ-function along the critical line