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Properties

Label 4-834-1.1-c1e2-0-0
Degree 44
Conductor 834834
Sign 11
Analytic cond. 0.05317650.0531765
Root an. cond. 0.4802080.480208
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  − 3-s − 4-s − 5-s − 3·7-s + 2·8-s − 2·9-s + 5·11-s + 12-s − 3·13-s + 15-s + 16-s − 2·17-s + 20-s + 3·21-s + 8·23-s − 2·24-s + 25-s + 2·27-s + 3·28-s + 29-s − 31-s − 4·32-s − 5·33-s + 3·35-s + 2·36-s − 4·37-s + 3·39-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.707·8-s − 2/3·9-s + 1.50·11-s + 0.288·12-s − 0.832·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.654·21-s + 1.66·23-s − 0.408·24-s + 1/5·25-s + 0.384·27-s + 0.566·28-s + 0.185·29-s − 0.179·31-s − 0.707·32-s − 0.870·33-s + 0.507·35-s + 1/3·36-s − 0.657·37-s + 0.480·39-s + ⋯

Functional equation

Λ(s)=(834s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(834s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 834834    =    2⋅3⋅1392 \cdot 3 \cdot 139
Sign: 11
Analytic conductor: 0.05317650.0531765
Root analytic conductor: 0.4802080.480208
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 834, ( :1/2,1/2), 1)(4,\ 834,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.36760988070.3676098807
L(12)L(\frac12) ≈\approx 0.36760988070.3676098807
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×\timesC2C_2 (1−T)(1+T+pT2) ( 1 - T )( 1 + T + p T^{2} )
3C1C_1×\timesC2C_2 (1+T)(1+pT2) ( 1 + T )( 1 + p T^{2} )
139C1C_1×\timesC2C_2 (1−T)(1−12T+pT2) ( 1 - T )( 1 - 12 T + p T^{2} )
good5D4D_{4} 1+T+pT3+p2T4 1 + T + p T^{3} + p^{2} T^{4}
7D4D_{4} 1+3T+6T2+3pT3+p2T4 1 + 3 T + 6 T^{2} + 3 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}
13D4D_{4} 1+3T+12T2+3pT3+p2T4 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+2T−6T2+2pT3+p2T4 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
29D4D_{4} 1−T−12T2−pT3+p2T4 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}
31D4D_{4} 1+T+14T2+pT3+p2T4 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
41D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+2T+46T2+2pT3+p2T4 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
59D4D_{4} 1−2T+62T2−2pT3+p2T4 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
67D4D_{4} 1+15T+186T2+15pT3+p2T4 1 + 15 T + 186 T^{2} + 15 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−3T+70T2−3pT3+p2T4 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )
79D4D_{4} 1−9T+30T2−9pT3+p2T4 1 - 9 T + 30 T^{2} - 9 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−7T+18T2−7pT3+p2T4 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+11T+112T2+11pT3+p2T4 1 + 11 T + 112 T^{2} + 11 p T^{3} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 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

−19.5498310312, −19.3435842630, −18.8747556104, −17.9344929515, −17.1949405684, −17.0187543082, −16.5276879710, −15.8754941792, −14.8988880324, −14.6818567971, −13.6718238204, −13.3213654465, −12.3808780905, −12.0170264206, −11.2183848234, −10.6046936283, −9.66599640236, −9.11838420690, −8.39429356341, −7.08082152627, −6.71343196046, −5.52246161490, −4.53472704076, −3.38079152731, 3.38079152731, 4.53472704076, 5.52246161490, 6.71343196046, 7.08082152627, 8.39429356341, 9.11838420690, 9.66599640236, 10.6046936283, 11.2183848234, 12.0170264206, 12.3808780905, 13.3213654465, 13.6718238204, 14.6818567971, 14.8988880324, 15.8754941792, 16.5276879710, 17.0187543082, 17.1949405684, 17.9344929515, 18.8747556104, 19.3435842630, 19.5498310312

Graph of the ZZ-function along the critical line