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Properties

Label 4-1110e2-1.1-c1e2-0-18
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 5-s − 6-s + 7-s − 8-s + 10-s + 12·11-s + 13-s + 14-s − 15-s − 16-s − 6·17-s + 7·19-s − 21-s + 12·22-s − 12·23-s + 24-s + 26-s + 27-s + 18·29-s − 30-s − 8·31-s − 12·33-s − 6·34-s + 35-s + 11·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 3.61·11-s + 0.277·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 1.60·19-s − 0.218·21-s + 2.55·22-s − 2.50·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 3.34·29-s − 0.182·30-s − 1.43·31-s − 2.08·33-s − 1.02·34-s + 0.169·35-s + 1.80·37-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 3.7784254313.778425431
L(12)L(\frac12) ≈\approx 3.7784254313.778425431
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3C2C_2 1+T+T2 1 + T + T^{2}
5C2C_2 1−T+T2 1 - T + T^{2}
37C2C_2 1−11T+pT2 1 - 11 T + p T^{2}
good7C2C_2 (1−5T+pT2)(1+4T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
13C22C_2^2 1−T−12T2−pT3+p2T4 1 - T - 12 T^{2} - 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}
19C2C_2 (1−8T+pT2)(1+T+pT2) ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 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}
43C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
47C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−6T−23T2−6pT3+p2T4 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+8T+3T2+8pT3+p2T4 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−10T+33T2−10pT3+p2T4 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−3T−62T2−3pT3+p2T4 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C22C_2^2 1+14T+117T2+14pT3+p2T4 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−3T−74T2−3pT3+p2T4 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−9.953871990082091940938851236081, −9.541927689555402402623443515630, −9.228378603034700844243454519974, −9.020732830746475835283969620751, −8.322048092597783534463145228437, −8.274585891953708611016338931558, −7.15395611354107512078693749426, −7.09264296644029755282107444231, −6.54609075013246584393561701979, −6.15886860605771446854277883946, −5.80399760247020061614618320794, −5.63168780548413473117476710773, −4.60502796469245766931162250838, −4.28906806490066852181619308797, −4.15348208551456601175354539965, −3.64903469629494670609858691602, −2.82786302841431761784719495373, −2.16321645654747343765217437596, −1.35268290978761980872789518326, −0.939947349923226685331139817729, 0.939947349923226685331139817729, 1.35268290978761980872789518326, 2.16321645654747343765217437596, 2.82786302841431761784719495373, 3.64903469629494670609858691602, 4.15348208551456601175354539965, 4.28906806490066852181619308797, 4.60502796469245766931162250838, 5.63168780548413473117476710773, 5.80399760247020061614618320794, 6.15886860605771446854277883946, 6.54609075013246584393561701979, 7.09264296644029755282107444231, 7.15395611354107512078693749426, 8.274585891953708611016338931558, 8.322048092597783534463145228437, 9.020732830746475835283969620751, 9.228378603034700844243454519974, 9.541927689555402402623443515630, 9.953871990082091940938851236081

Graph of the ZZ-function along the critical line