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Properties

Label 4-259200-1.1-c1e2-0-51
Degree 44
Conductor 259200259200
Sign −1-1
Analytic cond. 16.526816.5268
Root an. cond. 2.016262.01626
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 − 4·13-s − 4·17-s + 3·25-s + 4·29-s + 12·37-s + 12·41-s + 2·49-s − 12·53-s − 4·61-s + 8·65-s − 12·73-s + 8·85-s + 12·89-s − 28·97-s − 12·101-s + 28·109-s − 36·113-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.10·13-s − 0.970·17-s + 3/5·25-s + 0.742·29-s + 1.97·37-s + 1.87·41-s + 2/7·49-s − 1.64·53-s − 0.512·61-s + 0.992·65-s − 1.40·73-s + 0.867·85-s + 1.27·89-s − 2.84·97-s − 1.19·101-s + 2.68·109-s − 3.38·113-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

Λ(s)=(259200s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(259200s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 259200259200    =    27⋅34⋅522^{7} \cdot 3^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 16.526816.5268
Root analytic conductor: 2.016262.01626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 259200, ( :1/2,1/2), −1)(4,\ 259200,\ (\ :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
3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 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)2 ( 1 + 2 T + p T^{2} )^{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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1+14T+pT2)2 ( 1 + 14 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

−8.635268153066941217007271839152, −8.048390911320260435272830158835, −7.88125929411027787932176182764, −7.21857810559962191471917843249, −7.00520502565783528594752431202, −6.18178055218054403257438482099, −5.99573481729811170443753568761, −5.05802427658335260581937078682, −4.61878022728753276110930411726, −4.28002313553101117805216295071, −3.65788673520365604494122080438, −2.61772465980045468774058435809, −2.60538869357449517223391665122, −1.23519091396578725398334381678, 0, 1.23519091396578725398334381678, 2.60538869357449517223391665122, 2.61772465980045468774058435809, 3.65788673520365604494122080438, 4.28002313553101117805216295071, 4.61878022728753276110930411726, 5.05802427658335260581937078682, 5.99573481729811170443753568761, 6.18178055218054403257438482099, 7.00520502565783528594752431202, 7.21857810559962191471917843249, 7.88125929411027787932176182764, 8.048390911320260435272830158835, 8.635268153066941217007271839152

Graph of the ZZ-function along the critical line