↑

Properties

Label 4-234e2-1.1-c1e2-0-1
Degree 44
Conductor 5475654756
Sign 11
Analytic cond. 3.491293.49129
Root an. cond. 1.366931.36693
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 5-s + 3·6-s + 2·7-s + 8-s + 6·9-s + 10-s − 4·11-s − 7·13-s − 2·14-s + 3·15-s − 16-s − 7·17-s − 6·18-s − 19-s − 6·21-s + 4·22-s + 6·23-s − 3·24-s + 5·25-s + 7·26-s − 9·27-s − 6·29-s − 3·30-s + 7·31-s + 12·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 0.447·5-s + 1.22·6-s + 0.755·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s − 1.94·13-s − 0.534·14-s + 0.774·15-s − 1/4·16-s − 1.69·17-s − 1.41·18-s − 0.229·19-s − 1.30·21-s + 0.852·22-s + 1.25·23-s − 0.612·24-s + 25-s + 1.37·26-s − 1.73·27-s − 1.11·29-s − 0.547·30-s + 1.25·31-s + 2.08·33-s + ⋯

Functional equation

Λ(s)=(54756s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(54756s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5475654756    =    22⋅34⋅1322^{2} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3.491293.49129
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 54756, ( :1/2,1/2), 1)(4,\ 54756,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.25048355010.2504835501
L(12)L(\frac12) ≈\approx 0.25048355010.2504835501
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+pT+pT2 1 + p T + p T^{2}
13C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good5C22C_2^2 1+T−4T2+pT3+p2T4 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}
7C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C22C_2^2 1+4T+5T2+4pT3+p2T4 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+7T+32T2+7pT3+p2T4 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)(1+8T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} )
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 (1−11T+pT2)(1+4T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−10T+pT2)(1+11T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} )
41C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
43C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
47C22C_2^2 1+7T+2T2+7pT3+p2T4 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4}
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
67C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
71C22C_2^2 1−T−70T2−pT3+p2T4 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−17T+pT2)(1+4T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1+9T−2T2+9pT3+p2T4 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+7T−40T2+7pT3+p2T4 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4}
97C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
show more
show less
   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

−12.62950336776362834288227477311, −11.54384456155282791138014500674, −11.41921024566657346546370159419, −11.06118567233736727189840807078, −10.64559148883535576790675336170, −10.02763807828278873677062399261, −9.738516245441740608924730830309, −9.127621579876917035966212246415, −8.294237611889375955705362533278, −8.063066597226526063443247626508, −7.34149075361801390177133548258, −6.78377365433153306261818581578, −6.70635440531745501244315141001, −5.42447070190230526907917088766, −5.24356280733702663316269759244, −4.53130058097212535294355118558, −4.45669170973107871790583799204, −2.86502920266357870826712876930, −1.97517693997973803964703024685, −0.50158623033298012806731316187, 0.50158623033298012806731316187, 1.97517693997973803964703024685, 2.86502920266357870826712876930, 4.45669170973107871790583799204, 4.53130058097212535294355118558, 5.24356280733702663316269759244, 5.42447070190230526907917088766, 6.70635440531745501244315141001, 6.78377365433153306261818581578, 7.34149075361801390177133548258, 8.063066597226526063443247626508, 8.294237611889375955705362533278, 9.127621579876917035966212246415, 9.738516245441740608924730830309, 10.02763807828278873677062399261, 10.64559148883535576790675336170, 11.06118567233736727189840807078, 11.41921024566657346546370159419, 11.54384456155282791138014500674, 12.62950336776362834288227477311

Graph of the ZZ-function along the critical line