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Properties

Label 4-540800-1.1-c1e2-0-75
Degree 44
Conductor 540800540800
Sign −1-1
Analytic cond. 34.481834.4818
Root an. cond. 2.423242.42324
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·9-s + 2·13-s + 4·17-s + 3·25-s − 12·29-s − 12·37-s + 4·41-s − 4·45-s − 14·49-s − 4·53-s − 28·61-s + 4·65-s − 4·73-s − 5·81-s + 8·85-s − 12·89-s + 4·97-s − 20·101-s − 20·109-s + 4·113-s − 4·117-s − 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 0.554·13-s + 0.970·17-s + 3/5·25-s − 2.22·29-s − 1.97·37-s + 0.624·41-s − 0.596·45-s − 2·49-s − 0.549·53-s − 3.58·61-s + 0.496·65-s − 0.468·73-s − 5/9·81-s + 0.867·85-s − 1.27·89-s + 0.406·97-s − 1.99·101-s − 1.91·109-s + 0.376·113-s − 0.369·117-s − 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: −1-1
Analytic conductor: 34.481834.4818
Root analytic conductor: 2.423242.42324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 540800, ( :1/2,1/2), −1)(4,\ 540800,\ (\ :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)
bad2 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
13C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{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)2 ( 1 + 6 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

−8.282388148593036711889740506020, −7.76074024039613588419363915341, −7.44095337164983387150060505996, −6.79685282000922095191914374444, −6.33588991532526860429499225622, −5.89482538800335589915814344399, −5.43018572527764567927669449080, −5.24172374409154567149889570954, −4.42926641381854419615826380073, −3.83609522350488929158971696987, −3.10020205606080864107700760801, −2.93871105322865322021254198939, −1.66550824653592893392121562696, −1.63546887865200897104772810976, 0, 1.63546887865200897104772810976, 1.66550824653592893392121562696, 2.93871105322865322021254198939, 3.10020205606080864107700760801, 3.83609522350488929158971696987, 4.42926641381854419615826380073, 5.24172374409154567149889570954, 5.43018572527764567927669449080, 5.89482538800335589915814344399, 6.33588991532526860429499225622, 6.79685282000922095191914374444, 7.44095337164983387150060505996, 7.76074024039613588419363915341, 8.282388148593036711889740506020

Graph of the ZZ-function along the critical line