↑

Properties

Label 2-416-1.1-c3-0-8
Degree 22
Conductor 416416
Sign 11
Analytic cond. 24.544724.5447
Root an. cond. 4.954274.95427
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 5·7-s − 26·9-s − 10·11-s − 13·13-s + 15-s + 93·17-s + 82·19-s + 5·21-s + 192·23-s − 124·25-s + 53·27-s − 106·29-s − 172·31-s + 10·33-s + 5·35-s + 379·37-s + 13·39-s − 148·41-s + 329·43-s + 26·45-s + 631·47-s − 318·49-s − 93·51-s + 160·53-s + 10·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 0.0894·5-s − 0.269·7-s − 0.962·9-s − 0.274·11-s − 0.277·13-s + 0.0172·15-s + 1.32·17-s + 0.990·19-s + 0.0519·21-s + 1.74·23-s − 0.991·25-s + 0.377·27-s − 0.678·29-s − 0.996·31-s + 0.0527·33-s + 0.0241·35-s + 1.68·37-s + 0.0533·39-s − 0.563·41-s + 1.16·43-s + 0.0861·45-s + 1.95·47-s − 0.927·49-s − 0.255·51-s + 0.414·53-s + 0.0245·55-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 11
Analytic conductor: 24.544724.5447
Root analytic conductor: 4.954274.95427
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 416, ( :3/2), 1)(2,\ 416,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4790206731.479020673
L(12)L(\frac12) ≈\approx 1.4790206731.479020673
L(52)L(\frac{5}{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}
ppFp(T)F_p(T)
bad2 1 1
13 1+pT 1 + p T
good3 1+T+p3T2 1 + T + p^{3} T^{2}
5 1+T+p3T2 1 + T + p^{3} T^{2}
7 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
11 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
17 1−93T+p3T2 1 - 93 T + p^{3} T^{2}
19 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
23 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
29 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
31 1+172T+p3T2 1 + 172 T + p^{3} T^{2}
37 1−379T+p3T2 1 - 379 T + p^{3} T^{2}
41 1+148T+p3T2 1 + 148 T + p^{3} T^{2}
43 1−329T+p3T2 1 - 329 T + p^{3} T^{2}
47 1−631T+p3T2 1 - 631 T + p^{3} T^{2}
53 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
59 1−478T+p3T2 1 - 478 T + p^{3} T^{2}
61 1−300T+p3T2 1 - 300 T + p^{3} T^{2}
67 1−722T+p3T2 1 - 722 T + p^{3} T^{2}
71 1+335T+p3T2 1 + 335 T + p^{3} T^{2}
73 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
79 1−788T+p3T2 1 - 788 T + p^{3} T^{2}
83 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
89 1+866T+p3T2 1 + 866 T + p^{3} T^{2}
97 1+998T+p3T2 1 + 998 T + p^{3} T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.94380230604581157637976489044, −9.802567650855079415159037195789, −9.099782893744442444345235525495, −7.919908741852479378930667334438, −7.18043263731658985938298302545, −5.80194998140012414413386979161, −5.24451636886060418907270089159, −3.66637072604405912300150358104, −2.64129584064889962695520891714, −0.792207331854442004672296517207, 0.792207331854442004672296517207, 2.64129584064889962695520891714, 3.66637072604405912300150358104, 5.24451636886060418907270089159, 5.80194998140012414413386979161, 7.18043263731658985938298302545, 7.919908741852479378930667334438, 9.099782893744442444345235525495, 9.802567650855079415159037195789, 10.94380230604581157637976489044

Graph of the ZZ-function along the critical line