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Properties

Label 4-1500e2-1.1-c1e2-0-0
Degree 44
Conductor 22500002250000
Sign 11
Analytic cond. 143.461143.461
Root an. cond. 3.460863.46086
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 3·9-s + 2·11-s − 3·17-s + 5·19-s − 4·21-s − 3·23-s − 4·27-s + 8·29-s + 13·31-s − 4·33-s − 4·37-s + 2·43-s − 7·47-s − 6·49-s + 6·51-s − 53-s − 10·57-s + 18·59-s + 9·61-s + 6·63-s + 18·67-s + 6·69-s + 20·71-s + 16·73-s + 4·77-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 9-s + 0.603·11-s − 0.727·17-s + 1.14·19-s − 0.872·21-s − 0.625·23-s − 0.769·27-s + 1.48·29-s + 2.33·31-s − 0.696·33-s − 0.657·37-s + 0.304·43-s − 1.02·47-s − 6/7·49-s + 0.840·51-s − 0.137·53-s − 1.32·57-s + 2.34·59-s + 1.15·61-s + 0.755·63-s + 2.19·67-s + 0.722·69-s + 2.37·71-s + 1.87·73-s + 0.455·77-s + ⋯

Functional equation

Λ(s)=(2250000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2250000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2250000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2250000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 22500002250000    =    24⋅32⋅562^{4} \cdot 3^{2} \cdot 5^{6}
Sign: 11
Analytic conductor: 143.461143.461
Root analytic conductor: 3.460863.46086
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2250000, ( :1/2,1/2), 1)(4,\ 2250000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1175700292.117570029
L(12)L(\frac12) ≈\approx 2.1175700292.117570029
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
3C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
good7D4D_{4} 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 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}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17D4D_{4} 1+3T+35T2+3pT3+p2T4 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−5T+33T2−5pT3+p2T4 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+3T+37T2+3pT3+p2T4 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4}
29C4C_4 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−13T+103T2−13pT3+p2T4 1 - 13 T + 103 T^{2} - 13 p T^{3} + p^{2} T^{4}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
43D4D_{4} 1−2T+42T2−2pT3+p2T4 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+7T+75T2+7pT3+p2T4 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+T+75T2+pT3+p2T4 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4}
59D4D_{4} 1−18T+194T2−18pT3+p2T4 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−9T+41T2−9pT3+p2T4 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−18T+170T2−18pT3+p2T4 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−20T+222T2−20pT3+p2T4 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−16T+190T2−16pT3+p2T4 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−21T+257T2−21pT3+p2T4 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+3T+107T2+3pT3+p2T4 1 + 3 T + 107 T^{2} + 3 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+10T+198T2+10pT3+p2T4 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4}
97C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{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

−9.726432859737664682156228231807, −9.497005641017377038809873092866, −8.824577996080134723067605253424, −8.358961407170585384569609393986, −8.075526437938737313471231979071, −7.83928116943265167313604277616, −7.03070183816715098003664818653, −6.75246493362657133273706625553, −6.42152103991828550703463970755, −6.20275403311905355189319897979, −5.38905574574784813019059822265, −5.10094224524493417130218281167, −4.79421320658196128892458039315, −4.42362928754013325407308539896, −3.59172372259001658629139689784, −3.51687206381026694744303989327, −2.28290295872167970424742662503, −2.18517134564642826062018053617, −0.978694031598392741177382144865, −0.849676501480681629201491159721, 0.849676501480681629201491159721, 0.978694031598392741177382144865, 2.18517134564642826062018053617, 2.28290295872167970424742662503, 3.51687206381026694744303989327, 3.59172372259001658629139689784, 4.42362928754013325407308539896, 4.79421320658196128892458039315, 5.10094224524493417130218281167, 5.38905574574784813019059822265, 6.20275403311905355189319897979, 6.42152103991828550703463970755, 6.75246493362657133273706625553, 7.03070183816715098003664818653, 7.83928116943265167313604277616, 8.075526437938737313471231979071, 8.358961407170585384569609393986, 8.824577996080134723067605253424, 9.497005641017377038809873092866, 9.726432859737664682156228231807

Graph of the ZZ-function along the critical line