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Properties

Label 4-3200-1.1-c1e2-0-1
Degree 44
Conductor 32003200
Sign 11
Analytic cond. 0.2040340.204034
Root an. cond. 0.6720870.672087
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 + 2·7-s + 8-s − 9-s − 2·10-s + 5·11-s − 12-s − 6·13-s + 2·14-s + 2·15-s + 16-s − 3·17-s − 18-s − 3·19-s − 2·20-s − 2·21-s + 5·22-s − 2·23-s − 24-s − 25-s − 6·26-s + 2·28-s − 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 1/3·9-s − 0.632·10-s + 1.50·11-s − 0.288·12-s − 1.66·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.688·19-s − 0.447·20-s − 0.436·21-s + 1.06·22-s − 0.417·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.377·28-s − 0.371·29-s + ⋯

Functional equation

Λ(s)=(3200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 32003200    =    27⋅522^{7} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.2040340.204034
Root analytic conductor: 0.6720870.672087
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3200, ( :1/2,1/2), 1)(4,\ 3200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.84478043860.8447804386
L(12)L(\frac12) ≈\approx 0.84478043860.8447804386
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
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good3D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
7D4D_{4} 1−2T−2T2−2pT3+p2T4 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−5T+18T2−5pT3+p2T4 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+7T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} )
23D4D_{4} 1+2T+30T2+2pT3+p2T4 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−8T+pT2)(1+10T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )
31D4D_{4} 1+2T+30T2+2pT3+p2T4 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−10T+pT2)(1+3T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 3 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−12T+pT2)(1+4T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} )
47D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−12T+110T2−12pT3+p2T4 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−12T+pT2)(1+8T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )
61D4D_{4} 1−4T+78T2−4pT3+p2T4 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−9T+pT2)(1+12T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
73D4D_{4} 1+9T+148T2+9pT3+p2T4 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T+30T2−2pT3+p2T4 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+7T+58T2+7pT3+p2T4 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−T+104T2−pT3+p2T4 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+18T+pT2) ( 1 - 2 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

−17.8386888090, −17.5442935528, −17.0406118416, −16.6830275380, −16.0375643724, −15.3376859404, −14.8992914148, −14.4951797316, −14.0916295383, −13.2859500860, −12.4635176254, −12.1027752387, −11.6716606275, −11.2189813326, −10.6912421483, −9.76484458202, −9.02826212357, −8.31851072904, −7.39410638864, −7.07901565704, −6.06748159208, −5.38854611424, −4.30327828737, −4.06396508747, −2.35093004831, 2.35093004831, 4.06396508747, 4.30327828737, 5.38854611424, 6.06748159208, 7.07901565704, 7.39410638864, 8.31851072904, 9.02826212357, 9.76484458202, 10.6912421483, 11.2189813326, 11.6716606275, 12.1027752387, 12.4635176254, 13.2859500860, 14.0916295383, 14.4951797316, 14.8992914148, 15.3376859404, 16.0375643724, 16.6830275380, 17.0406118416, 17.5442935528, 17.8386888090

Graph of the ZZ-function along the critical line