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Properties

Label 4-86400-1.1-c1e2-0-3
Degree 44
Conductor 8640086400
Sign 11
Analytic cond. 5.508935.50893
Root an. cond. 1.532021.53202
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 12-s + 2·15-s + 16-s + 18-s − 2·20-s − 24-s − 25-s − 27-s + 2·30-s + 16·31-s + 32-s + 36-s − 2·40-s − 2·45-s − 48-s + 10·49-s − 50-s + 12·53-s − 54-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.447·20-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.365·30-s + 2.87·31-s + 0.176·32-s + 1/6·36-s − 0.316·40-s − 0.298·45-s − 0.144·48-s + 10/7·49-s − 0.141·50-s + 1.64·53-s − 0.136·54-s + ⋯

Functional equation

Λ(s)=(86400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(86400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8640086400    =    27⋅33⋅522^{7} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 5.508935.50893
Root analytic conductor: 1.532021.53202
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 86400, ( :1/2,1/2), 1)(4,\ 86400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6074717941.607471794
L(12)L(\frac12) ≈\approx 1.6074717941.607471794
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)
bad2C1C_1 1−T 1 - T
3C1C_1 1+T 1 + T
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p 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

−9.931431226499242282222146919527, −9.157336199204351077980875293537, −8.588983819760473461349159873067, −7.993325505837274428226208842001, −7.72383013837896400990817802991, −6.91860267058513733570889987370, −6.70671493958395410966482079315, −5.96241332719418166096565172640, −5.53507472959269894341516046375, −4.80161290199432400145705451230, −4.31416185438226392437568037299, −3.86701189083440126449303834685, −3.03842026071621200793542388091, −2.28949239380223024466903159190, −0.925919216123349724115980684505, 0.925919216123349724115980684505, 2.28949239380223024466903159190, 3.03842026071621200793542388091, 3.86701189083440126449303834685, 4.31416185438226392437568037299, 4.80161290199432400145705451230, 5.53507472959269894341516046375, 5.96241332719418166096565172640, 6.70671493958395410966482079315, 6.91860267058513733570889987370, 7.72383013837896400990817802991, 7.993325505837274428226208842001, 8.588983819760473461349159873067, 9.157336199204351077980875293537, 9.931431226499242282222146919527

Graph of the ZZ-function along the critical line