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Properties

Label 4-588e2-1.1-c1e2-0-35
Degree 44
Conductor 345744345744
Sign −1-1
Analytic cond. 22.044922.0449
Root an. cond. 2.166842.16684
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  + 3-s − 2·9-s + 4·13-s − 2·19-s − 25-s − 5·27-s − 14·31-s − 2·37-s + 4·39-s − 8·43-s − 2·57-s − 2·61-s − 14·67-s − 2·73-s − 75-s − 26·79-s + 81-s − 14·93-s − 20·97-s + 22·103-s − 2·109-s − 2·111-s − 8·117-s − 13·121-s + 127-s − 8·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.10·13-s − 0.458·19-s − 1/5·25-s − 0.962·27-s − 2.51·31-s − 0.328·37-s + 0.640·39-s − 1.21·43-s − 0.264·57-s − 0.256·61-s − 1.71·67-s − 0.234·73-s − 0.115·75-s − 2.92·79-s + 1/9·81-s − 1.45·93-s − 2.03·97-s + 2.16·103-s − 0.191·109-s − 0.189·111-s − 0.739·117-s − 1.18·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(345744s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(345744s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345744345744    =    24⋅32⋅742^{4} \cdot 3^{2} \cdot 7^{4}
Sign: −1-1
Analytic conductor: 22.044922.0449
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 345744, ( :1/2,1/2), −1)(4,\ 345744,\ (\ :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)
bad2 1 1
3C2C_2 1−T+pT2 1 - T + p T^{2}
7 1 1
good5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + 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+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
37C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
59C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
79C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−15T+pT2)(1+15T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )
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

−8.646784595072977282306137209646, −8.262637018939911096603858847029, −7.51155146227211557820915407148, −7.27657735328498763814943158038, −6.67413798525527807153917855846, −6.02455306491682974275355684196, −5.71711213824960626307579454428, −5.28434328079604079625535316421, −4.44529829557613531132250015524, −3.98660375345473223094112675622, −3.31407617989174490462667281240, −3.04451713874417289677681835886, −2.04448974040096779312937543400, −1.54505867526772913791014367907, 0, 1.54505867526772913791014367907, 2.04448974040096779312937543400, 3.04451713874417289677681835886, 3.31407617989174490462667281240, 3.98660375345473223094112675622, 4.44529829557613531132250015524, 5.28434328079604079625535316421, 5.71711213824960626307579454428, 6.02455306491682974275355684196, 6.67413798525527807153917855846, 7.27657735328498763814943158038, 7.51155146227211557820915407148, 8.262637018939911096603858847029, 8.646784595072977282306137209646

Graph of the ZZ-function along the critical line