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Properties

Label 4-855e2-1.1-c1e2-0-3
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5-s − 4·7-s + 4·8-s − 2·10-s + 6·11-s − 6·13-s − 8·14-s + 8·16-s + 6·17-s − 7·19-s − 2·20-s + 12·22-s − 8·23-s − 12·26-s − 8·28-s + 7·29-s + 18·31-s + 8·32-s + 12·34-s + 4·35-s − 4·37-s − 14·38-s − 4·40-s + 6·41-s − 10·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 1.51·7-s + 1.41·8-s − 0.632·10-s + 1.80·11-s − 1.66·13-s − 2.13·14-s + 2·16-s + 1.45·17-s − 1.60·19-s − 0.447·20-s + 2.55·22-s − 1.66·23-s − 2.35·26-s − 1.51·28-s + 1.29·29-s + 3.23·31-s + 1.41·32-s + 2.05·34-s + 0.676·35-s − 0.657·37-s − 2.27·38-s − 0.632·40-s + 0.937·41-s − 1.52·43-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6705983043.670598304
L(12)L(\frac12) ≈\approx 3.6705983043.670598304
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)
bad3 1 1
5C2C_2 1+T+T2 1 + T + T^{2}
19C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good2C22C_2^2 1−pT+pT2−p2T3+p2T4 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4}
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C22C_2^2 1+6T+23T2+6pT3+p2T4 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + 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}
31C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1−6T−5T2−6pT3+p2T4 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+10T+57T2+10pT3+p2T4 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−4T−31T2−4pT3+p2T4 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−14T+143T2−14pT3+p2T4 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−3T−50T2−3pT3+p2T4 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−7T−12T2−7pT3+p2T4 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+4T−51T2+4pT3+p2T4 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−7T−22T2−7pT3+p2T4 1 - 7 T - 22 T^{2} - 7 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+5T−54T2+5pT3+p2T4 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89C22C_2^2 1−3T−80T2−3pT3+p2T4 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−12T+47T2−12pT3+p2T4 1 - 12 T + 47 T^{2} - 12 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

−10.21702140489915585562847361904, −10.21429459004235498656816916133, −9.654284415219037582720301206542, −9.390251081839685706395585325617, −8.510665612064594147786901559654, −8.180833630540973111928522988921, −7.85639046269681270025750241220, −7.16421258992752142002467748583, −6.71686569055411375209606613324, −6.36809279174628169687719555083, −6.28299973055076573697172599269, −5.39677435190006361182924611538, −5.04283324769869745509954827452, −4.22206591085346042349226903022, −4.21509924737431123894137942729, −3.79106148424710745346216876567, −3.03081996372827882612646759374, −2.65123854997023272671893607167, −1.78497635569571174038079952676, −0.76184847244173038283792271090, 0.76184847244173038283792271090, 1.78497635569571174038079952676, 2.65123854997023272671893607167, 3.03081996372827882612646759374, 3.79106148424710745346216876567, 4.21509924737431123894137942729, 4.22206591085346042349226903022, 5.04283324769869745509954827452, 5.39677435190006361182924611538, 6.28299973055076573697172599269, 6.36809279174628169687719555083, 6.71686569055411375209606613324, 7.16421258992752142002467748583, 7.85639046269681270025750241220, 8.180833630540973111928522988921, 8.510665612064594147786901559654, 9.390251081839685706395585325617, 9.654284415219037582720301206542, 10.21429459004235498656816916133, 10.21702140489915585562847361904

Graph of the ZZ-function along the critical line