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Properties

Label 4-86400-1.1-c1e2-0-20
Degree 44
Conductor 8640086400
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 4·5-s − 6-s − 8-s + 9-s + 4·10-s + 12-s − 4·15-s + 16-s − 18-s − 8·19-s − 4·20-s + 2·23-s − 24-s + 11·25-s + 27-s + 10·29-s + 4·30-s − 32-s + 36-s + 8·38-s + 4·40-s − 2·43-s − 4·45-s − 2·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.894·20-s + 0.417·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s + 1.85·29-s + 0.730·30-s − 0.176·32-s + 1/6·36-s + 1.29·38-s + 0.632·40-s − 0.304·43-s − 0.596·45-s − 0.294·46-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: −1-1
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: 11
Selberg data: (4, 86400, ( :1/2,1/2), −1)(4,\ 86400,\ (\ :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)
bad2C1C_1 1+T 1 + T
3C1C_1 1−T 1 - T
5C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good7C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
13C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−8T+pT2)(1−2T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )
31C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
37C22C_2^2 1−24T2+p2T4 1 - 24 T^{2} + p^{2} T^{4}
41C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−6T+pT2)(1+8T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+pT2)(1+12T+pT2) ( 1 + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
61C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+8T+pT2)(1+12T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+pT2) ( 1 - 10 T + p T^{2} )( 1 + p T^{2} )
79C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
83C22C_2^2 1+122T2+p2T4 1 + 122 T^{2} + p^{2} T^{4}
89C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+10T+pT2)(1+12T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 12 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

−9.339567424682786430784453915247, −8.673694079790885226677238103851, −8.362799533606162582050264738401, −8.159133907905646093204816135301, −7.60740765702098772208639847156, −6.99699840001464631994095197982, −6.59540810342012874523421779116, −6.11652543717961444714848438316, −4.81772000759950662932311022970, −4.63656510970371918111347666082, −3.88273063652071506820021780492, −3.20886576284072801053693960150, −2.66417888904161937297868600496, −1.46767016993003419922605384526, 0, 1.46767016993003419922605384526, 2.66417888904161937297868600496, 3.20886576284072801053693960150, 3.88273063652071506820021780492, 4.63656510970371918111347666082, 4.81772000759950662932311022970, 6.11652543717961444714848438316, 6.59540810342012874523421779116, 6.99699840001464631994095197982, 7.60740765702098772208639847156, 8.159133907905646093204816135301, 8.362799533606162582050264738401, 8.673694079790885226677238103851, 9.339567424682786430784453915247

Graph of the ZZ-function along the critical line