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Properties

Label 4-240e2-1.1-c1e2-0-27
Degree 44
Conductor 5760057600
Sign −1-1
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
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  − 2·4-s − 4·7-s + 9-s + 2·11-s − 2·13-s + 4·16-s + 4·17-s − 4·19-s + 8·23-s − 5·25-s + 8·28-s − 6·29-s + 2·31-s − 2·36-s − 10·37-s − 4·41-s − 12·43-s − 4·44-s + 8·47-s + 2·49-s + 4·52-s − 2·59-s − 14·61-s − 4·63-s − 8·64-s − 4·67-s − 8·68-s + ⋯
L(s)  = 1  − 4-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 16-s + 0.970·17-s − 0.917·19-s + 1.66·23-s − 25-s + 1.51·28-s − 1.11·29-s + 0.359·31-s − 1/3·36-s − 1.64·37-s − 0.624·41-s − 1.82·43-s − 0.603·44-s + 1.16·47-s + 2/7·49-s + 0.554·52-s − 0.260·59-s − 1.79·61-s − 0.503·63-s − 64-s − 0.488·67-s − 0.970·68-s + ⋯

Functional equation

Λ(s)=(57600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(57600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5760057600    =    28⋅32⋅522^{8} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 57600, ( :1/2,1/2), −1)(4,\ 57600,\ (\ :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)
bad2C2C_2 1+pT2 1 + p T^{2}
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C2C_2 1+pT2 1 + p T^{2}
good7C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2×\timesC2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
13D4D_{4} 1+2T+6T2+2pT3+p2T4 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
23D4D_{4} 1−8T+38T2−8pT3+p2T4 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−2T−6T2−2pT3+p2T4 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+10T+78T2+10pT3+p2T4 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4}
41C4C_4 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+12T+102T2+12pT3+p2T4 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−8T+62T2−8pT3+p2T4 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4}
53C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−12T+pT2)(1+14T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} )
61D4D_{4} 1+14T+114T2+14pT3+p2T4 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
73D4D_{4} 1−10T+114T2−10pT3+p2T4 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T−38T2−2pT3+p2T4 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
89C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
97D4D_{4} 1+6T+138T2+6pT3+p2T4 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4}
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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

−14.8595826501, −14.3820983435, −13.7410216765, −13.4512544705, −13.1203440067, −12.5766943432, −12.2093864102, −11.9594843409, −11.0713887673, −10.5604845975, −10.0406635436, −9.69164586099, −9.34365210340, −8.81028299029, −8.40099932774, −7.54559136120, −7.21584134111, −6.48002006446, −6.13128908222, −5.26829567139, −4.91780353436, −3.97563996350, −3.54080710729, −2.96031601073, −1.57454212811, 0, 1.57454212811, 2.96031601073, 3.54080710729, 3.97563996350, 4.91780353436, 5.26829567139, 6.13128908222, 6.48002006446, 7.21584134111, 7.54559136120, 8.40099932774, 8.81028299029, 9.34365210340, 9.69164586099, 10.0406635436, 10.5604845975, 11.0713887673, 11.9594843409, 12.2093864102, 12.5766943432, 13.1203440067, 13.4512544705, 13.7410216765, 14.3820983435, 14.8595826501

Graph of the ZZ-function along the critical line