Properties

Label 16-5e32-1.1-c1e8-0-10
Degree $16$
Conductor $2.328\times 10^{22}$
Sign $1$
Analytic cond. $384819.$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 10·9-s − 4·11-s − 5·13-s − 16-s + 15·17-s + 10·19-s + 15·23-s − 15·27-s + 15·29-s + 31-s + 20·33-s − 5·37-s + 25·39-s − 9·41-s − 15·47-s + 5·48-s + 35·49-s − 75·51-s + 35·53-s − 50·57-s + 15·59-s + 6·61-s − 5·64-s − 75·69-s − 29·71-s + 10·73-s + ⋯
L(s)  = 1  − 2.88·3-s + 10/3·9-s − 1.20·11-s − 1.38·13-s − 1/4·16-s + 3.63·17-s + 2.29·19-s + 3.12·23-s − 2.88·27-s + 2.78·29-s + 0.179·31-s + 3.48·33-s − 0.821·37-s + 4.00·39-s − 1.40·41-s − 2.18·47-s + 0.721·48-s + 5·49-s − 10.5·51-s + 4.80·53-s − 6.62·57-s + 1.95·59-s + 0.768·61-s − 5/8·64-s − 9.02·69-s − 3.44·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{32}\)
Sign: $1$
Analytic conductor: \(384819.\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.455501360\)
\(L(\frac12)\) \(\approx\) \(2.455501360\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + T^{4} + 5 T^{6} + 5 T^{7} - 9 T^{8} + 5 p T^{9} + 5 p^{2} T^{10} + p^{4} T^{12} + p^{8} T^{16} \)
3 \( 1 + 5 T + 5 p T^{2} + 40 T^{3} + 32 p T^{4} + 205 T^{5} + 430 T^{6} + 850 T^{7} + 1531 T^{8} + 850 p T^{9} + 430 p^{2} T^{10} + 205 p^{3} T^{11} + 32 p^{5} T^{12} + 40 p^{5} T^{13} + 5 p^{7} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - 5 p T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 7045 p^{2} T^{10} + 611 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 36 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 5 T + 40 T^{2} + 200 T^{3} + 956 T^{4} + 4155 T^{5} + 15835 T^{6} + 61940 T^{7} + 217296 T^{8} + 61940 p T^{9} + 15835 p^{2} T^{10} + 4155 p^{3} T^{11} + 956 p^{4} T^{12} + 200 p^{5} T^{13} + 40 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 15 T + 120 T^{2} - 570 T^{3} + 1786 T^{4} - 4065 T^{5} + 17855 T^{6} - 123410 T^{7} + 652356 T^{8} - 123410 p T^{9} + 17855 p^{2} T^{10} - 4065 p^{3} T^{11} + 1786 p^{4} T^{12} - 570 p^{5} T^{13} + 120 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 10 T + 12 T^{2} + 150 T^{3} - 197 T^{4} - 1210 T^{5} - 5166 T^{6} + 23900 T^{7} + 31275 T^{8} + 23900 p T^{9} - 5166 p^{2} T^{10} - 1210 p^{3} T^{11} - 197 p^{4} T^{12} + 150 p^{5} T^{13} + 12 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 15 T + 155 T^{2} - 1270 T^{3} + 8906 T^{4} - 55565 T^{5} + 314590 T^{6} - 1661560 T^{7} + 8221031 T^{8} - 1661560 p T^{9} + 314590 p^{2} T^{10} - 55565 p^{3} T^{11} + 8906 p^{4} T^{12} - 1270 p^{5} T^{13} + 155 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 15 T + 92 T^{2} - 380 T^{3} + 1868 T^{4} - 5065 T^{5} - 16861 T^{6} + 71400 T^{7} + 212600 T^{8} + 71400 p T^{9} - 16861 p^{2} T^{10} - 5065 p^{3} T^{11} + 1868 p^{4} T^{12} - 380 p^{5} T^{13} + 92 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - T - 65 T^{2} - 150 T^{3} + 860 T^{4} + 15887 T^{5} + 66068 T^{6} - 319870 T^{7} - 3557825 T^{8} - 319870 p T^{9} + 66068 p^{2} T^{10} + 15887 p^{3} T^{11} + 860 p^{4} T^{12} - 150 p^{5} T^{13} - 65 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 5 T + 75 T^{2} - 360 T^{3} - 994 T^{4} - 42345 T^{5} + 70860 T^{6} - 186740 T^{7} + 11831221 T^{8} - 186740 p T^{9} + 70860 p^{2} T^{10} - 42345 p^{3} T^{11} - 994 p^{4} T^{12} - 360 p^{5} T^{13} + 75 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 9 T - 20 T^{2} - 520 T^{3} - 1320 T^{4} + 22927 T^{5} + 153103 T^{6} - 256460 T^{7} - 5290960 T^{8} - 256460 p T^{9} + 153103 p^{2} T^{10} + 22927 p^{3} T^{11} - 1320 p^{4} T^{12} - 520 p^{5} T^{13} - 20 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 5 p T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
47 \( 1 + 15 T + 120 T^{2} + 960 T^{3} + 9636 T^{4} + 80865 T^{5} + 573845 T^{6} + 3818880 T^{7} + 26386476 T^{8} + 3818880 p T^{9} + 573845 p^{2} T^{10} + 80865 p^{3} T^{11} + 9636 p^{4} T^{12} + 960 p^{5} T^{13} + 120 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 35 T + 655 T^{2} - 8020 T^{3} + 67266 T^{4} - 345785 T^{5} + 173700 T^{6} + 16131620 T^{7} - 171061659 T^{8} + 16131620 p T^{9} + 173700 p^{2} T^{10} - 345785 p^{3} T^{11} + 67266 p^{4} T^{12} - 8020 p^{5} T^{13} + 655 p^{6} T^{14} - 35 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 15 T + 102 T^{2} - 840 T^{3} + 9468 T^{4} - 65115 T^{5} + 401879 T^{6} - 4632450 T^{7} + 44690100 T^{8} - 4632450 p T^{9} + 401879 p^{2} T^{10} - 65115 p^{3} T^{11} + 9468 p^{4} T^{12} - 840 p^{5} T^{13} + 102 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 6 T + 130 T^{2} - 135 T^{3} + 11205 T^{4} + 15672 T^{5} + 635993 T^{6} + 3243465 T^{7} + 39830270 T^{8} + 3243465 p T^{9} + 635993 p^{2} T^{10} + 15672 p^{3} T^{11} + 11205 p^{4} T^{12} - 135 p^{5} T^{13} + 130 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 130 T^{2} - 460 T^{3} + 6651 T^{4} - 59800 T^{5} + 247100 T^{6} - 6229960 T^{7} + 6681701 T^{8} - 6229960 p T^{9} + 247100 p^{2} T^{10} - 59800 p^{3} T^{11} + 6651 p^{4} T^{12} - 460 p^{5} T^{13} + 130 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 29 T + 320 T^{2} + 1480 T^{3} - 1630 T^{4} - 56363 T^{5} - 3597 T^{6} + 7019210 T^{7} + 88563540 T^{8} + 7019210 p T^{9} - 3597 p^{2} T^{10} - 56363 p^{3} T^{11} - 1630 p^{4} T^{12} + 1480 p^{5} T^{13} + 320 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 10 T + 220 T^{2} - 2525 T^{3} + 28471 T^{4} - 303710 T^{5} + 2569015 T^{6} - 24707575 T^{7} + 202533026 T^{8} - 24707575 p T^{9} + 2569015 p^{2} T^{10} - 303710 p^{3} T^{11} + 28471 p^{4} T^{12} - 2525 p^{5} T^{13} + 220 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 10 T + 112 T^{2} - 20 T^{3} - 2177 T^{4} - 102490 T^{5} - 216986 T^{6} - 1018300 T^{7} + 47464475 T^{8} - 1018300 p T^{9} - 216986 p^{2} T^{10} - 102490 p^{3} T^{11} - 2177 p^{4} T^{12} - 20 p^{5} T^{13} + 112 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 15 T + 155 T^{2} + 60 T^{3} - 11364 T^{4} + 155385 T^{5} - 467570 T^{6} - 4330590 T^{7} + 97045931 T^{8} - 4330590 p T^{9} - 467570 p^{2} T^{10} + 155385 p^{3} T^{11} - 11364 p^{4} T^{12} + 60 p^{5} T^{13} + 155 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 40 T + 562 T^{2} - 2590 T^{3} - 7557 T^{4} + 41760 T^{5} + 1856844 T^{6} - 44856200 T^{7} + 566687025 T^{8} - 44856200 p T^{9} + 1856844 p^{2} T^{10} + 41760 p^{3} T^{11} - 7557 p^{4} T^{12} - 2590 p^{5} T^{13} + 562 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 10 T + 75 T^{2} + 1560 T^{3} + 3371 T^{4} - 6540 T^{5} + 92385 T^{6} - 7411990 T^{7} - 78500144 T^{8} - 7411990 p T^{9} + 92385 p^{2} T^{10} - 6540 p^{3} T^{11} + 3371 p^{4} T^{12} + 1560 p^{5} T^{13} + 75 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.88246670727134565260815334959, −4.65961356074932702076863814770, −4.50806995417856037073059889501, −4.48828139473731481697998454802, −3.95756110055778982264517968620, −3.83465381180779367684435392312, −3.75430245919133842705565509700, −3.75185613413115827505057453672, −3.50407881407620518036839902784, −3.31179702025238202596461018028, −3.24395101422803210159536885261, −3.03614822265795697209152811876, −2.83743101365522621841638051095, −2.81925597549133803027735284493, −2.62146289288296814702160782195, −2.32528922951762497084568107271, −2.16079881493560758230595424895, −2.08697595013944376760068630204, −1.86382862361576112001884538024, −1.27648093192640814676643534957, −1.10421441499899767923854050169, −1.01558198900896374371762983638, −0.835232834462728783299950408826, −0.56598992628096162920116903435, −0.55167843555714431159086209040, 0.55167843555714431159086209040, 0.56598992628096162920116903435, 0.835232834462728783299950408826, 1.01558198900896374371762983638, 1.10421441499899767923854050169, 1.27648093192640814676643534957, 1.86382862361576112001884538024, 2.08697595013944376760068630204, 2.16079881493560758230595424895, 2.32528922951762497084568107271, 2.62146289288296814702160782195, 2.81925597549133803027735284493, 2.83743101365522621841638051095, 3.03614822265795697209152811876, 3.24395101422803210159536885261, 3.31179702025238202596461018028, 3.50407881407620518036839902784, 3.75185613413115827505057453672, 3.75430245919133842705565509700, 3.83465381180779367684435392312, 3.95756110055778982264517968620, 4.48828139473731481697998454802, 4.50806995417856037073059889501, 4.65961356074932702076863814770, 4.88246670727134565260815334959

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.