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Properties

Label 4-43904-1.1-c1e2-0-10
Degree 44
Conductor 4390443904
Sign −1-1
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
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 + 4-s − 2·5-s + 7-s − 8-s − 2·9-s + 2·10-s + 2·13-s − 14-s + 16-s − 6·17-s + 2·18-s − 2·20-s − 8·23-s + 2·25-s − 2·26-s + 28-s − 4·29-s + 4·31-s − 32-s + 6·34-s − 2·35-s − 2·36-s + 4·37-s + 2·40-s + 2·41-s − 16·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.447·20-s − 1.66·23-s + 2/5·25-s − 0.392·26-s + 0.188·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/3·36-s + 0.657·37-s + 0.316·40-s + 0.312·41-s − 2.43·43-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :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
7C1C_1 1−T 1 - T
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+6T+26T2+6pT3+p2T4 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
29D4D_{4} 1+4T+14T2+4pT3+p2T4 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+14T2−4pT3+p2T4 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+16T+126T2+16pT3+p2T4 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+4T−34T2+4pT3+p2T4 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−4T+pT2)(1+14T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} )
67D4D_{4} 1+4T+70T2+4pT3+p2T4 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73D4D_{4} 1−2T−14T2−2pT3+p2T4 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+78T2+8pT3+p2T4 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−16T+162T2−16pT3+p2T4 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+pT2)(1+14T+pT2) ( 1 + p T^{2} )( 1 + 14 T + p T^{2} )
97D4D_{4} 1−2T−70T2−2pT3+p2T4 1 - 2 T - 70 T^{2} - 2 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

−15.2044887106, −14.7986331807, −14.2654895574, −13.7901197244, −13.2108905524, −12.9095825673, −11.9955195911, −11.8060432483, −11.3859674867, −11.0619071903, −10.4231508360, −10.0151603446, −9.30317318374, −8.83273717856, −8.27406452484, −8.08512942671, −7.50843087012, −6.74347751107, −6.34129777909, −5.70964625786, −4.86898551946, −4.19718886633, −3.57300806656, −2.65335980391, −1.72691978313, 0, 1.72691978313, 2.65335980391, 3.57300806656, 4.19718886633, 4.86898551946, 5.70964625786, 6.34129777909, 6.74347751107, 7.50843087012, 8.08512942671, 8.27406452484, 8.83273717856, 9.30317318374, 10.0151603446, 10.4231508360, 11.0619071903, 11.3859674867, 11.8060432483, 11.9955195911, 12.9095825673, 13.2108905524, 13.7901197244, 14.2654895574, 14.7986331807, 15.2044887106

Graph of the ZZ-function along the critical line