↑

Properties

Label 2-416-1.1-c3-0-28
Degree 22
Conductor 416416
Sign −1-1
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 11

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: −1-1
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: 11
Selberg data: (2, 416, ( :3/2), −1)(2,\ 416,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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

−10.20046068172938889162196910891, −9.539285530984600710001055192192, −8.214198244785336280501904603421, −7.930640773995339376148046080683, −6.40857065564106316434350429548, −5.61613784608317344015897061709, −4.32945773792707444165859889356, −3.17442755373337081861436461936, −1.84086539159772154942608010769, 0, 1.84086539159772154942608010769, 3.17442755373337081861436461936, 4.32945773792707444165859889356, 5.61613784608317344015897061709, 6.40857065564106316434350429548, 7.930640773995339376148046080683, 8.214198244785336280501904603421, 9.539285530984600710001055192192, 10.20046068172938889162196910891

Graph of the ZZ-function along the critical line