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Properties

Label 4-540e2-1.1-c0e2-0-1
Degree 44
Conductor 291600291600
Sign 11
Analytic cond. 0.07262760.0726276
Root an. cond. 0.5191290.519129
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 + 5-s + 7-s − 8-s + 10-s + 14-s − 16-s − 23-s − 29-s + 35-s − 40-s − 41-s − 2·43-s − 46-s − 47-s + 49-s − 56-s − 58-s + 61-s + 64-s + 67-s + 70-s − 80-s − 82-s − 83-s − 2·86-s + 2·89-s + ⋯
L(s)  = 1  + 2-s + 5-s + 7-s − 8-s + 10-s + 14-s − 16-s − 23-s − 29-s + 35-s − 40-s − 41-s − 2·43-s − 46-s − 47-s + 49-s − 56-s − 58-s + 61-s + 64-s + 67-s + 70-s − 80-s − 82-s − 83-s − 2·86-s + 2·89-s + ⋯

Functional equation

Λ(s)=(291600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(291600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 291600291600    =    24⋅36⋅522^{4} \cdot 3^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.07262760.0726276
Root analytic conductor: 0.5191290.519129
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 291600, ( :0,0), 1)(4,\ 291600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3622403441.362240344
L(12)L(\frac12) ≈\approx 1.3622403441.362240344
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}
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C1C_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)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
29C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + 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}
41C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
43C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
47C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + 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)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
89C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{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

−11.70225816721088247338415274469, −10.89664684612161506203859970544, −10.23325456326375212320291222532, −10.05384256778273470565226863028, −9.499566059348226330290444137282, −9.146436524116860180669508754909, −8.478473002340239398692614497072, −8.277525887460268212554830754048, −7.70815261552396702511844814822, −7.06821756023737041524021857298, −6.44199065028002407760298267795, −6.18358575565569073566325391242, −5.52900259488754075856356566529, −5.06249212530784345061425221832, −4.97919979098446550649991846714, −4.01063383425975005748374740452, −3.75325010947034271814718782800, −2.93819880043402743187568511838, −2.12142809824458251968121301207, −1.65891548502932067179509079584, 1.65891548502932067179509079584, 2.12142809824458251968121301207, 2.93819880043402743187568511838, 3.75325010947034271814718782800, 4.01063383425975005748374740452, 4.97919979098446550649991846714, 5.06249212530784345061425221832, 5.52900259488754075856356566529, 6.18358575565569073566325391242, 6.44199065028002407760298267795, 7.06821756023737041524021857298, 7.70815261552396702511844814822, 8.277525887460268212554830754048, 8.478473002340239398692614497072, 9.146436524116860180669508754909, 9.499566059348226330290444137282, 10.05384256778273470565226863028, 10.23325456326375212320291222532, 10.89664684612161506203859970544, 11.70225816721088247338415274469

Graph of the ZZ-function along the critical line