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Properties

Label 4-504e2-1.1-c0e2-0-0
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 0.06326670.0632667
Root an. cond. 0.5015260.501526
Motivic weight 00
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  − 2-s − 2·3-s − 5-s + 2·6-s − 7-s + 8-s + 3·9-s + 10-s + 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s + 23-s − 2·24-s + 25-s − 2·26-s − 4·27-s − 2·30-s + 35-s − 2·38-s − 4·39-s − 40-s − 2·42-s − 3·45-s + ⋯
L(s)  = 1  − 2-s − 2·3-s − 5-s + 2·6-s − 7-s + 8-s + 3·9-s + 10-s + 2·13-s + 14-s + 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s + 23-s − 2·24-s + 25-s − 2·26-s − 4·27-s − 2·30-s + 35-s − 2·38-s − 4·39-s − 40-s − 2·42-s − 3·45-s + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.06326670.0632667
Root analytic conductor: 0.5015260.501526
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 254016, ( :0,0), 1)(4,\ 254016,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.19209831800.1920983180
L(12)L(\frac12) ≈\approx 0.19209831800.1920983180
L(1)L(1) 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)
bad2C2C_2 1+T+T2 1 + T + T^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
7C2C_2 1+T+T2 1 + T + T^{2}
good5C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
23C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
61C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
83C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{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

−11.53807098707566870139151734339, −10.84714600543848557037994163102, −10.42221649036897612790423638392, −10.30356323187396066346087723628, −9.596514782753109281686101974481, −9.182627850665603609226991013567, −8.914048376752615677525062219855, −8.145240894306979817949908777756, −7.61066734956533661391529047058, −7.37309771885186120907951493767, −6.67027968540024341763391217447, −6.51818548568021431076483180974, −5.86682746942423310554254338530, −5.17581826413679925562563802003, −5.01052437586629966409980113166, −4.03627033499675448547696490701, −3.82233050635133160964287254791, −3.09464739201220127694230549691, −1.38591687451132591132221097641, −0.881950195852900869598048203446, 0.881950195852900869598048203446, 1.38591687451132591132221097641, 3.09464739201220127694230549691, 3.82233050635133160964287254791, 4.03627033499675448547696490701, 5.01052437586629966409980113166, 5.17581826413679925562563802003, 5.86682746942423310554254338530, 6.51818548568021431076483180974, 6.67027968540024341763391217447, 7.37309771885186120907951493767, 7.61066734956533661391529047058, 8.145240894306979817949908777756, 8.914048376752615677525062219855, 9.182627850665603609226991013567, 9.596514782753109281686101974481, 10.30356323187396066346087723628, 10.42221649036897612790423638392, 10.84714600543848557037994163102, 11.53807098707566870139151734339

Graph of the ZZ-function along the critical line