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Properties

Label 4-91e2-1.1-c1e2-0-5
Degree 44
Conductor 82818281
Sign 11
Analytic cond. 0.5280030.528003
Root an. cond. 0.8524310.852431
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 6·3-s + 2·4-s − 3·5-s + 6·6-s − 5·7-s − 5·8-s + 21·9-s + 3·10-s − 6·11-s − 12·12-s − 2·13-s + 5·14-s + 18·15-s + 5·16-s + 2·17-s − 21·18-s − 2·19-s − 6·20-s + 30·21-s + 6·22-s + 30·24-s + 5·25-s + 2·26-s − 54·27-s − 10·28-s − 7·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 3.46·3-s + 4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.76·8-s + 7·9-s + 0.948·10-s − 1.80·11-s − 3.46·12-s − 0.554·13-s + 1.33·14-s + 4.64·15-s + 5/4·16-s + 0.485·17-s − 4.94·18-s − 0.458·19-s − 1.34·20-s + 6.54·21-s + 1.27·22-s + 6.12·24-s + 25-s + 0.392·26-s − 10.3·27-s − 1.88·28-s − 1.29·29-s + ⋯

Functional equation

Λ(s)=(8281s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8281s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 82818281    =    72⋅1327^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.5280030.528003
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 8281, ( :1/2,1/2), 1)(4,\ 8281,\ (\ :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)
bad7C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
13C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good2C22C_2^2 1+T−T2+pT3+p2T4 1 + T - T^{2} + p T^{3} + p^{2} T^{4}
3C2C_2 (1+pT+pT2)2 ( 1 + p T + p T^{2} )^{2}
5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
17C22C_2^2 1−2T−13T2−2pT3+p2T4 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1+7T+20T2+7pT3+p2T4 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+3T−22T2+3pT3+p2T4 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+2T−33T2+2pT3+p2T4 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+3T−32T2+3pT3+p2T4 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−7T+6T2−7pT3+p2T4 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+T−46T2+pT3+p2T4 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4}
53C22C_2^2 1+3T−44T2+3pT3+p2T4 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
67C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
71C22C_2^2 1+13T+98T2+13pT3+p2T4 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−13T+96T2−13pT3+p2T4 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−3T−70T2−3pT3+p2T4 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+14T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 14 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

−13.17083072304890003935458042810, −12.87148941030843382217326914293, −12.35600926309536103741998797111, −12.08027994337324273952628204747, −11.82021533416337375398473062326, −11.03561694622216597655332642963, −10.61928492568430797996677418294, −10.54540591994848998103598344623, −9.710119850407055439006879556857, −9.180931829567947556896723749772, −7.74588984405759713913977021044, −7.37774941531169181062342851270, −6.70721235102570303894165474320, −6.33600266034391586792357201625, −5.59003972084021475716683567771, −5.38223469062928345016302182981, −4.19143906619076892162692347148, −3.04189453050271718744975784464, 0, 0, 3.04189453050271718744975784464, 4.19143906619076892162692347148, 5.38223469062928345016302182981, 5.59003972084021475716683567771, 6.33600266034391586792357201625, 6.70721235102570303894165474320, 7.37774941531169181062342851270, 7.74588984405759713913977021044, 9.180931829567947556896723749772, 9.710119850407055439006879556857, 10.54540591994848998103598344623, 10.61928492568430797996677418294, 11.03561694622216597655332642963, 11.82021533416337375398473062326, 12.08027994337324273952628204747, 12.35600926309536103741998797111, 12.87148941030843382217326914293, 13.17083072304890003935458042810

Graph of the ZZ-function along the critical line