↑

Properties

Label 4-5408e2-1.1-c1e2-0-9
Degree 44
Conductor 2924646429246464
Sign 11
Analytic cond. 1864.771864.77
Root an. cond. 6.571386.57138
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  + 2·3-s + 6·7-s − 9-s − 2·11-s − 6·17-s + 6·19-s + 12·21-s + 6·23-s − 2·25-s − 6·27-s + 6·29-s − 4·33-s − 2·37-s + 6·41-s + 6·43-s + 12·47-s + 15·49-s − 12·51-s + 12·57-s + 6·59-s + 14·61-s − 6·63-s + 18·67-s + 12·69-s − 6·71-s − 8·73-s − 4·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.26·7-s − 1/3·9-s − 0.603·11-s − 1.45·17-s + 1.37·19-s + 2.61·21-s + 1.25·23-s − 2/5·25-s − 1.15·27-s + 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.937·41-s + 0.914·43-s + 1.75·47-s + 15/7·49-s − 1.68·51-s + 1.58·57-s + 0.781·59-s + 1.79·61-s − 0.755·63-s + 2.19·67-s + 1.44·69-s − 0.712·71-s − 0.936·73-s − 0.461·75-s + ⋯

Functional equation

Λ(s)=(29246464s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29246464s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2924646429246464    =    210⋅1342^{10} \cdot 13^{4}
Sign: 11
Analytic conductor: 1864.771864.77
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 29246464, ( :1/2,1/2), 1)(4,\ 29246464,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.2275899397.227589939
L(12)L(\frac12) ≈\approx 7.2275899397.227589939
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
13 1 1
good3D4D_{4} 1−2T+5T2−2pT3+p2T4 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4}
5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
7D4D_{4} 1−6T+3pT2−6pT3+p2T4 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+2T+5T2+2pT3+p2T4 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+6T+35T2+6pT3+p2T4 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−6T+29T2−6pT3+p2T4 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−6T+37T2−6pT3+p2T4 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−6T+35T2−6pT3+p2T4 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
37D4D_{4} 1+2T+3T2+2pT3+p2T4 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+83T2−6pT3+p2T4 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−6T+45T2−6pT3+p2T4 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4}
47C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
53C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
59D4D_{4} 1−6T+109T2−6pT3+p2T4 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
67D4D_{4} 1−18T+197T2−18pT3+p2T4 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+6T+133T2+6pT3+p2T4 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+8T+90T2+8pT3+p2T4 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4}
79C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89D4D_{4} 1−18T+227T2−18pT3+p2T4 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4}
97C2C_2 (1−9T+pT2)2 ( 1 - 9 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

−8.163002021581106903165300470532, −8.114993285590380313870147274227, −7.80260847911718824346767896848, −7.48768716252550825910324545140, −7.02670816124336143730025666275, −6.74815102104003767634057697284, −6.21158818162114220661713603136, −5.60232411800284578314719073859, −5.41867426713687951873961939835, −4.99160731173212377682020100826, −4.80414533850404788822905637947, −4.31453885098004132106142828463, −3.87484245944889210098693645298, −3.49568209900732479762338730748, −2.76205241891822037752605317653, −2.68208283939971958074764602206, −2.05182273091076588194428043791, −2.00228685098563934553756299625, −0.968780946651762960778269978958, −0.789429202330035618389570570369, 0.789429202330035618389570570369, 0.968780946651762960778269978958, 2.00228685098563934553756299625, 2.05182273091076588194428043791, 2.68208283939971958074764602206, 2.76205241891822037752605317653, 3.49568209900732479762338730748, 3.87484245944889210098693645298, 4.31453885098004132106142828463, 4.80414533850404788822905637947, 4.99160731173212377682020100826, 5.41867426713687951873961939835, 5.60232411800284578314719073859, 6.21158818162114220661713603136, 6.74815102104003767634057697284, 7.02670816124336143730025666275, 7.48768716252550825910324545140, 7.80260847911718824346767896848, 8.114993285590380313870147274227, 8.163002021581106903165300470532

Graph of the ZZ-function along the critical line