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Properties

Label 4-1110e2-1.1-c1e2-0-7
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 + 4·7-s + 8-s − 10-s + 6·11-s + 13-s − 4·14-s − 15-s − 16-s − 3·17-s − 2·19-s − 4·21-s − 6·22-s − 24-s − 26-s + 27-s − 6·29-s + 30-s − 8·31-s − 6·33-s + 3·34-s + 4·35-s + 11·37-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.277·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.872·21-s − 1.27·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s − 1.04·33-s + 0.514·34-s + 0.676·35-s + 1.80·37-s + 0.324·38-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 1.5763116951.576311695
L(12)L(\frac12) ≈\approx 1.5763116951.576311695
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+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−3T+pT2)2 ( 1 - 3 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+3T−8T2+3pT3+p2T4 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
53C22C_2^2 1−pT2+p2T4 1 - p T^{2} + 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 1−10T+39T2−10pT3+p2T4 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−13T+102T2−13pT3+p2T4 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+6T−35T2+6pT3+p2T4 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4}
73C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
79C22C_2^2 1−16T+177T2−16pT3+p2T4 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+12T+61T2+12pT3+p2T4 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−6T−53T2−6pT3+p2T4 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−9.878239054423569226636079936166, −9.633258673698703187860387889141, −9.065505156518732485211837234604, −8.999582932397830252761232864753, −8.508513689942342705592719183176, −8.051834098891151457851135003076, −7.56782466759933503910621492650, −7.29661070892775548685309680603, −6.65102638732774285269019993368, −6.21966732912253373981346406212, −6.00643948418209814195783956982, −5.27649957827327823292054789467, −4.99890206292190340404723554584, −4.26007425008852788319433041974, −4.12805427966953877989478370948, −3.51204786983622105353654680162, −2.46758968551662027939843231709, −1.84743324634711795347959393023, −1.49088098446221964131153324016, −0.69945685765039033126907425969, 0.69945685765039033126907425969, 1.49088098446221964131153324016, 1.84743324634711795347959393023, 2.46758968551662027939843231709, 3.51204786983622105353654680162, 4.12805427966953877989478370948, 4.26007425008852788319433041974, 4.99890206292190340404723554584, 5.27649957827327823292054789467, 6.00643948418209814195783956982, 6.21966732912253373981346406212, 6.65102638732774285269019993368, 7.29661070892775548685309680603, 7.56782466759933503910621492650, 8.051834098891151457851135003076, 8.508513689942342705592719183176, 8.999582932397830252761232864753, 9.065505156518732485211837234604, 9.633258673698703187860387889141, 9.878239054423569226636079936166

Graph of the ZZ-function along the critical line