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Properties

Label 4-1110e2-1.1-c1e2-0-1
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  − 2·3-s − 4-s + 4·7-s + 3·9-s − 8·11-s + 2·12-s + 16-s − 8·21-s − 25-s − 4·27-s − 4·28-s + 16·33-s − 3·36-s − 12·37-s − 4·41-s + 8·44-s − 12·47-s − 2·48-s − 2·49-s − 8·53-s + 12·63-s − 64-s − 24·67-s + 24·71-s − 20·73-s + 2·75-s − 32·77-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 1.51·7-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s − 1.74·21-s − 1/5·25-s − 0.769·27-s − 0.755·28-s + 2.78·33-s − 1/2·36-s − 1.97·37-s − 0.624·41-s + 1.20·44-s − 1.75·47-s − 0.288·48-s − 2/7·49-s − 1.09·53-s + 1.51·63-s − 1/8·64-s − 2.93·67-s + 2.84·71-s − 2.34·73-s + 0.230·75-s − 3.64·77-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 0.39339039960.3933903996
L(12)L(\frac12) ≈\approx 0.39339039960.3933903996
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+T2 1 + T^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
5C2C_2 1+T2 1 + T^{2}
37C2C_2 1+12T+pT2 1 + 12 T + p T^{2}
good7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
53C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
59C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
61C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
97C22C_2^2 1−158T2+p2T4 1 - 158 T^{2} + 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

−10.25233601242330338090564003971, −9.549344045755351280133870684840, −9.540082499143076009899418847400, −8.606803320091890227767562289421, −8.275500262423017108059551941863, −7.941092806026474074300442167043, −7.82373939850816971140897808123, −6.97706187661899317846342614308, −6.90922549752020937169875704460, −6.08364303326375243717514241185, −5.58482274861421329117222928399, −5.19831875646926909265083324229, −5.10598899958593967280850762312, −4.55093287159720382558945460654, −4.24763845043794295734265975699, −3.23654449668267964420466352171, −2.91158535905479085598150828743, −1.72754324329536463598166408976, −1.70132295377990458461839531936, −0.29542925061326266326070552891, 0.29542925061326266326070552891, 1.70132295377990458461839531936, 1.72754324329536463598166408976, 2.91158535905479085598150828743, 3.23654449668267964420466352171, 4.24763845043794295734265975699, 4.55093287159720382558945460654, 5.10598899958593967280850762312, 5.19831875646926909265083324229, 5.58482274861421329117222928399, 6.08364303326375243717514241185, 6.90922549752020937169875704460, 6.97706187661899317846342614308, 7.82373939850816971140897808123, 7.941092806026474074300442167043, 8.275500262423017108059551941863, 8.606803320091890227767562289421, 9.540082499143076009899418847400, 9.549344045755351280133870684840, 10.25233601242330338090564003971

Graph of the ZZ-function along the critical line