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Properties

Label 4-1110e2-1.1-c1e2-0-23
Degree 44
Conductor 12321001232100
Sign −1-1
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 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 2·7-s + 9-s + 2·12-s + 16-s − 12·19-s + 4·21-s + 25-s − 4·27-s + 2·28-s + 6·31-s + 36-s − 2·37-s − 2·43-s + 2·48-s − 11·49-s − 24·57-s − 30·61-s + 2·63-s + 64-s + 2·75-s − 12·76-s − 16·79-s − 11·81-s + 4·84-s + 12·93-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.577·12-s + 1/4·16-s − 2.75·19-s + 0.872·21-s + 1/5·25-s − 0.769·27-s + 0.377·28-s + 1.07·31-s + 1/6·36-s − 0.328·37-s − 0.304·43-s + 0.288·48-s − 1.57·49-s − 3.17·57-s − 3.84·61-s + 0.251·63-s + 1/8·64-s + 0.230·75-s − 1.37·76-s − 1.80·79-s − 1.22·81-s + 0.436·84-s + 1.24·93-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: −1-1
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: 11
Selberg data: (4, 1232100, ( :1/2,1/2), −1)(4,\ 1232100,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
37C1C_1 (1+T)2 ( 1 + T )^{2}
good7C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
29C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
31C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−13T+pT2)(1+13T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
73C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
97C2C_2 (1+7T+pT2)2 ( 1 + 7 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

−7.947480595907964918208604648618, −7.55947946358028496449425566831, −6.89931657836489782864864420449, −6.56405990058932844390013504676, −6.08056229568137165407273954903, −5.77272193873314481326296907642, −4.83332190202465636941309325541, −4.63919100492136925221940798677, −4.12586421650220620613319696777, −3.56154188220255317605428378165, −2.82505147340032154850850367219, −2.65455170884079945670816110578, −1.73957054101312727200784638700, −1.63805431218632083963465992264, 0, 1.63805431218632083963465992264, 1.73957054101312727200784638700, 2.65455170884079945670816110578, 2.82505147340032154850850367219, 3.56154188220255317605428378165, 4.12586421650220620613319696777, 4.63919100492136925221940798677, 4.83332190202465636941309325541, 5.77272193873314481326296907642, 6.08056229568137165407273954903, 6.56405990058932844390013504676, 6.89931657836489782864864420449, 7.55947946358028496449425566831, 7.947480595907964918208604648618

Graph of the ZZ-function along the critical line