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Properties

Label 4-117e2-1.1-c1e2-0-14
Degree 44
Conductor 1368913689
Sign 11
Analytic cond. 0.8728220.872822
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 3·8-s + 6·9-s + 9·10-s − 6·11-s − 12·12-s − 5·13-s + 9·15-s + 3·16-s − 3·17-s − 18·18-s − 3·19-s − 12·20-s + 18·22-s − 6·23-s + 9·24-s + 25-s + 15·26-s − 9·27-s + 6·29-s − 27·30-s − 15·31-s − 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 1.06·8-s + 2·9-s + 2.84·10-s − 1.80·11-s − 3.46·12-s − 1.38·13-s + 2.32·15-s + 3/4·16-s − 0.727·17-s − 4.24·18-s − 0.688·19-s − 2.68·20-s + 3.83·22-s − 1.25·23-s + 1.83·24-s + 1/5·25-s + 2.94·26-s − 1.73·27-s + 1.11·29-s − 4.92·30-s − 2.69·31-s − 1.06·32-s + ⋯

Functional equation

Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(13689s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1368913689    =    34⋅1323^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.8728220.872822
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 13689, ( :1/2,1/2), 1)(4,\ 13689,\ (\ :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)
bad3C2C_2 1+pT+pT2 1 + p T + p T^{2}
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good2C22C_2^2 1+3T+5T2+3pT3+p2T4 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}
5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
11C22C_2^2 1+6T+23T2+6pT3+p2T4 1 + 6 T + 23 T^{2} + 6 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+3T+22T2+3pT3+p2T4 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
23C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
29C22C_2^2 1−6T+7T2−6pT3+p2T4 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4}
31C2C_2 (1+4T+pT2)(1+11T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C2C_2 (1−T+pT2)(1+10T+pT2) ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} )
41C22C_2^2 1+65T2+p2T4 1 + 65 T^{2} + p^{2} T^{4}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1+9T+74T2+9pT3+p2T4 1 + 9 T + 74 T^{2} + 9 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−6T+71T2−6pT3+p2T4 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
67C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
71C22C_2^2 1+15T+146T2+15pT3+p2T4 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
79C22C_2^2 1−11T+42T2−11pT3+p2T4 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+9T+110T2+9pT3+p2T4 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−27T+332T2−27pT3+p2T4 1 - 27 T + 332 T^{2} - 27 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+49T2+p2T4 1 + 49 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

−12.98354886104095708706834147321, −12.35165226491345955772984980588, −12.03435267932437668437033047366, −11.66807984831276045420323313624, −10.84897546948938748494242978061, −10.56959718220958168531791237660, −10.27377683308262605034854737643, −9.780559707535456130152025476849, −8.923219890284389932905477106741, −8.480946020083334206735078351896, −7.72713435356032831754083300337, −7.47458363107256614223360335633, −7.03439552964069623066767208973, −6.05908241654551215166689785058, −5.29878391325248697301060944214, −4.75082974210359934621131928119, −3.79158084802677213002512374174, −2.15276262389819621973419206531, 0, 0, 2.15276262389819621973419206531, 3.79158084802677213002512374174, 4.75082974210359934621131928119, 5.29878391325248697301060944214, 6.05908241654551215166689785058, 7.03439552964069623066767208973, 7.47458363107256614223360335633, 7.72713435356032831754083300337, 8.480946020083334206735078351896, 8.923219890284389932905477106741, 9.780559707535456130152025476849, 10.27377683308262605034854737643, 10.56959718220958168531791237660, 10.84897546948938748494242978061, 11.66807984831276045420323313624, 12.03435267932437668437033047366, 12.35165226491345955772984980588, 12.98354886104095708706834147321

Graph of the ZZ-function along the critical line