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Properties

Label 4-115200-1.1-c1e2-0-11
Degree 44
Conductor 115200115200
Sign 11
Analytic cond. 7.345257.34525
Root an. cond. 1.646271.64627
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  + 8·7-s + 9-s − 4·17-s − 16·23-s + 25-s + 20·41-s + 16·47-s + 34·49-s + 8·63-s − 28·73-s + 32·79-s + 81-s + 4·89-s + 4·97-s + 8·103-s + 12·113-s − 32·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s − 128·161-s + ⋯
L(s)  = 1  + 3.02·7-s + 1/3·9-s − 0.970·17-s − 3.33·23-s + 1/5·25-s + 3.12·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s − 3.27·73-s + 3.60·79-s + 1/9·81-s + 0.423·89-s + 0.406·97-s + 0.788·103-s + 1.12·113-s − 2.93·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s − 10.0·161-s + ⋯

Functional equation

Λ(s)=(115200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(115200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 115200115200    =    29⋅32⋅522^{9} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 7.345257.34525
Root analytic conductor: 1.646271.64627
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 115200, ( :1/2,1/2), 1)(4,\ 115200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1917499752.191749975
L(12)L(\frac12) ≈\approx 2.1917499752.191749975
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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+8T+pT2)2 ( 1 + 8 T + 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+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{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)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 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 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−9.138788409882082860623313179234, −9.082691646086291194533627292052, −8.338450200049268291048738447687, −7.929044973464791963060046214031, −7.64493796343622540106320647343, −7.34594365176835478558310775458, −6.25442273107448540439252397130, −5.92809906508794084871962623137, −5.28897888504504330389281711769, −4.67467612086321652791764027956, −4.13728859524865050162716547657, −4.05477376880357436694530570270, −2.25919867640796924053383456153, −2.19180563382997996757520036314, −1.20192952395638838673519576448, 1.20192952395638838673519576448, 2.19180563382997996757520036314, 2.25919867640796924053383456153, 4.05477376880357436694530570270, 4.13728859524865050162716547657, 4.67467612086321652791764027956, 5.28897888504504330389281711769, 5.92809906508794084871962623137, 6.25442273107448540439252397130, 7.34594365176835478558310775458, 7.64493796343622540106320647343, 7.929044973464791963060046214031, 8.338450200049268291048738447687, 9.082691646086291194533627292052, 9.138788409882082860623313179234

Graph of the ZZ-function along the critical line