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Properties

Label 4-5000e2-1.1-c1e2-0-2
Degree 44
Conductor 2500000025000000
Sign 11
Analytic cond. 1594.021594.02
Root an. cond. 6.318636.31863
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  + 3·7-s − 9-s − 2·11-s + 9·13-s + 6·17-s + 13·19-s + 2·23-s + 11·29-s + 4·31-s − 12·37-s − 2·41-s + 11·43-s + 7·47-s − 6·49-s − 2·53-s + 7·59-s − 4·61-s − 3·63-s + 14·67-s − 15·71-s + 22·73-s − 6·77-s + 5·79-s − 8·81-s − 18·83-s + 8·89-s + 27·91-s + ⋯
L(s)  = 1  + 1.13·7-s − 1/3·9-s − 0.603·11-s + 2.49·13-s + 1.45·17-s + 2.98·19-s + 0.417·23-s + 2.04·29-s + 0.718·31-s − 1.97·37-s − 0.312·41-s + 1.67·43-s + 1.02·47-s − 6/7·49-s − 0.274·53-s + 0.911·59-s − 0.512·61-s − 0.377·63-s + 1.71·67-s − 1.78·71-s + 2.57·73-s − 0.683·77-s + 0.562·79-s − 8/9·81-s − 1.97·83-s + 0.847·89-s + 2.83·91-s + ⋯

Functional equation

Λ(s)=(25000000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 25000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(25000000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2500000025000000    =    26⋅582^{6} \cdot 5^{8}
Sign: 11
Analytic conductor: 1594.021594.02
Root analytic conductor: 6.318636.31863
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25000000, ( :1/2,1/2), 1)(4,\ 25000000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.1053342616.105334261
L(12)L(\frac12) ≈\approx 6.1053342616.105334261
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)
bad2 1 1
5 1 1
good3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7D4D_{4} 1−3T+15T2−3pT3+p2T4 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+2T+18T2+2pT3+p2T4 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−9T+45T2−9pT3+p2T4 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−6T+38T2−6pT3+p2T4 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C4C_4 1−13T+79T2−13pT3+p2T4 1 - 13 T + 79 T^{2} - 13 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−2T+27T2−2pT3+p2T4 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−11T+3pT2−11pT3+p2T4 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+21T2−4pT3+p2T4 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+12T+105T2+12pT3+p2T4 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+2T+38T2+2pT3+p2T4 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−11T+115T2−11pT3+p2T4 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−7T+95T2−7pT3+p2T4 1 - 7 T + 95 T^{2} - 7 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+2T+27T2+2pT3+p2T4 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−7T+129T2−7pT3+p2T4 1 - 7 T + 129 T^{2} - 7 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+4T+T2+4pT3+p2T4 1 + 4 T + T^{2} + 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−14T+138T2−14pT3+p2T4 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+15T+197T2+15pT3+p2T4 1 + 15 T + 197 T^{2} + 15 p T^{3} + p^{2} T^{4}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79D4D_{4} 1−5T+103T2−5pT3+p2T4 1 - 5 T + 103 T^{2} - 5 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+18T+227T2+18pT3+p2T4 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4}
89C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
97D4D_{4} 1−3T−85T2−3pT3+p2T4 1 - 3 T - 85 T^{2} - 3 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

−8.192660890612915937822827089075, −8.171468694813778809144105003464, −7.906789520717673940418250839956, −7.39545509050943832331015684352, −6.93989234591975969654859890725, −6.80169174494682962693753143714, −6.09304979740468563550205765905, −5.82163544502398333233585073069, −5.40321432622224000307388270604, −5.31193522432327444491710009300, −4.85700885409352141610756859134, −4.39466907416120587769116232784, −3.82990876918081890920588906452, −3.45365841968313447796631580677, −2.99440252639100394478674746036, −2.97055251605350105672258357243, −2.07009313886406504464342892506, −1.40296079073813550058691134081, −0.990239120071540947013169629800, −0.925563979991074371201302994640, 0.925563979991074371201302994640, 0.990239120071540947013169629800, 1.40296079073813550058691134081, 2.07009313886406504464342892506, 2.97055251605350105672258357243, 2.99440252639100394478674746036, 3.45365841968313447796631580677, 3.82990876918081890920588906452, 4.39466907416120587769116232784, 4.85700885409352141610756859134, 5.31193522432327444491710009300, 5.40321432622224000307388270604, 5.82163544502398333233585073069, 6.09304979740468563550205765905, 6.80169174494682962693753143714, 6.93989234591975969654859890725, 7.39545509050943832331015684352, 7.906789520717673940418250839956, 8.171468694813778809144105003464, 8.192660890612915937822827089075

Graph of the ZZ-function along the critical line