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Properties

Label 4-1110e2-1.1-c1e2-0-9
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
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 − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s − 4·10-s + 3·11-s − 6·12-s − 3·13-s + 2·14-s − 4·15-s + 5·16-s + 3·17-s − 6·18-s + 3·19-s + 6·20-s + 2·21-s − 6·22-s + 23-s + 8·24-s + 3·25-s + 6·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.904·11-s − 1.73·12-s − 0.832·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.727·17-s − 1.41·18-s + 0.688·19-s + 1.34·20-s + 0.436·21-s − 1.27·22-s + 0.208·23-s + 1.63·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.98788453380.9878845338
L(12)L(\frac12) ≈\approx 0.98788453380.9878845338
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 (1+T)2 ( 1 + T )^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
5C1C_1 (1−T)2 ( 1 - T )^{2}
37C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1+T+6T2+pT3+p2T4 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4}
11D4D_{4} 1−3T+16T2−3pT3+p2T4 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+3T+20T2+3pT3+p2T4 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−3T+28T2−3pT3+p2T4 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−3T+32T2−3pT3+p2T4 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−T+38T2−pT3+p2T4 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31D4D_{4} 1+6T+38T2+6pT3+p2T4 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+58T2−6pT3+p2T4 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47D4D_{4} 1−2T+62T2−2pT3+p2T4 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−5T+104T2−5pT3+p2T4 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−18T+166T2−18pT3+p2T4 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−6T+98T2−6pT3+p2T4 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+6T+110T2+6pT3+p2T4 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−14T+158T2−14pT3+p2T4 1 - 14 T + 158 T^{2} - 14 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−9T+92T2−9pT3+p2T4 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4}
79C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
83D4D_{4} 1−7T+170T2−7pT3+p2T4 1 - 7 T + 170 T^{2} - 7 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−21T+280T2−21pT3+p2T4 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−4T+66T2−4pT3+p2T4 1 - 4 T + 66 T^{2} - 4 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

−9.949991477259471093589151146249, −9.785166473869818352386352040370, −9.233962082308994120919370484186, −9.132533672252278719758903061035, −8.362721335182838771891454882484, −8.166292991905687590382797175022, −7.37109103034187218877082252533, −7.19383089129740938666023607805, −6.64367677864087529475494925552, −6.49574926978295521754769688108, −5.91323595865813664960378916141, −5.58163714663183829670662499015, −4.89680318143745927342182143807, −4.78377900172501744495823396513, −3.57733667604527650710703646354, −3.41648934671731639952703452985, −2.35880976293082522865229911219, −2.06079048662367972003274439679, −1.04556555540668435101019840246, −0.75820356854681945346810287386, 0.75820356854681945346810287386, 1.04556555540668435101019840246, 2.06079048662367972003274439679, 2.35880976293082522865229911219, 3.41648934671731639952703452985, 3.57733667604527650710703646354, 4.78377900172501744495823396513, 4.89680318143745927342182143807, 5.58163714663183829670662499015, 5.91323595865813664960378916141, 6.49574926978295521754769688108, 6.64367677864087529475494925552, 7.19383089129740938666023607805, 7.37109103034187218877082252533, 8.166292991905687590382797175022, 8.362721335182838771891454882484, 9.132533672252278719758903061035, 9.233962082308994120919370484186, 9.785166473869818352386352040370, 9.949991477259471093589151146249

Graph of the ZZ-function along the critical line