Properties

Label 16-150e8-1.1-c10e8-0-0
Degree $16$
Conductor $2.563\times 10^{17}$
Sign $1$
Analytic cond. $6.80571\times 10^{15}$
Root an. cond. $9.76235$
Motivic weight $10$
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  + 2.04e3·4-s − 1.59e5·9-s + 2.62e6·16-s + 3.13e6·19-s − 4.35e7·31-s − 3.25e8·36-s + 1.17e9·49-s − 2.36e9·61-s + 2.68e9·64-s + 6.42e9·76-s − 3.98e8·79-s + 1.33e10·81-s + 4.06e10·109-s + 9.08e10·121-s − 8.92e10·124-s + 127-s + 131-s + 137-s + 139-s − 4.16e11·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.55e11·169-s − 4.98e11·171-s + ⋯
L(s)  = 1  + 2·4-s − 2.69·9-s + 5/2·16-s + 1.26·19-s − 1.52·31-s − 5.38·36-s + 4.17·49-s − 2.80·61-s + 5/2·64-s + 2.53·76-s − 0.129·79-s + 3.83·81-s + 2.64·109-s + 3.50·121-s − 3.04·124-s − 6.73·144-s + 6.20·169-s − 3.41·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(6.80571\times 10^{15}\)
Root analytic conductor: \(9.76235\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{150} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [5]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.302428442\)
\(L(\frac12)\) \(\approx\) \(2.302428442\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{9} T^{2} )^{4} \)
3 \( 1 + 17668 p^{2} T^{2} + 1815718 p^{8} T^{4} + 17668 p^{22} T^{6} + p^{40} T^{8} \)
5 \( 1 \)
good7 \( ( 1 - 589979708 T^{2} + 3550354063112742 p^{2} T^{4} - 589979708 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
11 \( ( 1 - 4130589740 p T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - 4130589740 p^{21} T^{6} + p^{40} T^{8} )^{2} \)
13 \( ( 1 - 427750447196 T^{2} + \)\(82\!\cdots\!06\)\( T^{4} - 427750447196 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
17 \( ( 1 + 503008419460 T^{2} - \)\(28\!\cdots\!38\)\( T^{4} + 503008419460 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
19 \( ( 1 - 784364 T + 11072641146486 T^{2} - 784364 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
23 \( ( 1 + 132650537376580 T^{2} + \)\(14\!\cdots\!78\)\( p^{2} T^{4} + 132650537376580 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
29 \( ( 1 - 775524833463844 T^{2} + \)\(31\!\cdots\!86\)\( T^{4} - 775524833463844 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
31 \( ( 1 + 10892924 T + 345177991175046 T^{2} + 10892924 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
37 \( ( 1 - 9378946365819548 T^{2} + \)\(62\!\cdots\!18\)\( T^{4} - 9378946365819548 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
41 \( ( 1 - 30129232679351620 T^{2} + \)\(47\!\cdots\!42\)\( T^{4} - 30129232679351620 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
43 \( ( 1 - 45615259605803804 T^{2} + \)\(10\!\cdots\!46\)\( T^{4} - 45615259605803804 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
47 \( ( 1 + 185271765343089796 T^{2} + \)\(13\!\cdots\!06\)\( T^{4} + 185271765343089796 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
53 \( ( 1 + 486913415004520420 T^{2} + \)\(10\!\cdots\!62\)\( T^{4} + 486913415004520420 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
59 \( ( 1 - 1720402455795118180 T^{2} + \)\(12\!\cdots\!62\)\( T^{4} - 1720402455795118180 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
61 \( ( 1 + 592019372 T + 1459215526973846838 T^{2} + 592019372 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
67 \( ( 1 - 6568829083389764828 T^{2} + \)\(17\!\cdots\!58\)\( T^{4} - 6568829083389764828 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
71 \( ( 1 - 11173485987747195844 T^{2} + \)\(52\!\cdots\!86\)\( T^{4} - 11173485987747195844 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
73 \( ( 1 - 11243329304249234876 T^{2} + \)\(65\!\cdots\!46\)\( T^{4} - 11243329304249234876 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
79 \( ( 1 + 99641284 T + 15267587176773126726 T^{2} + 99641284 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
83 \( ( 1 + 56087025146752445092 T^{2} + \)\(12\!\cdots\!58\)\( T^{4} + 56087025146752445092 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
89 \( ( 1 - 97642754408129363140 T^{2} + \)\(41\!\cdots\!62\)\( T^{4} - 97642754408129363140 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
97 \( ( 1 - \)\(10\!\cdots\!64\)\( T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!64\)\( p^{20} T^{6} + p^{40} T^{8} )^{2} \)
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

−4.11170392903757222581133334658, −3.81540857071461933515989233314, −3.60481549083769462291188564087, −3.42446971051215239225600529119, −3.32265376748223940441548832178, −3.11502646547986903447376276696, −3.10695673882521244508692296655, −3.08634921290407597368939515914, −2.75333119558039016847702495297, −2.59083109891444334535273636167, −2.44347211532622590679067670518, −2.38692124945801935214171434196, −2.07312220039832839593369890261, −1.99067428269361222439880123551, −1.95742752735735317643691040819, −1.72337831717102526372073017489, −1.44982910169287158763740065236, −1.38765361595016762637237892355, −0.988602187253098440295486836816, −0.895370877168620173456499341852, −0.869234711952092063443326529547, −0.73145358007294026746120474456, −0.35827142772740589500670650743, −0.30201616462484960846975386877, −0.07249611009353107557283541923, 0.07249611009353107557283541923, 0.30201616462484960846975386877, 0.35827142772740589500670650743, 0.73145358007294026746120474456, 0.869234711952092063443326529547, 0.895370877168620173456499341852, 0.988602187253098440295486836816, 1.38765361595016762637237892355, 1.44982910169287158763740065236, 1.72337831717102526372073017489, 1.95742752735735317643691040819, 1.99067428269361222439880123551, 2.07312220039832839593369890261, 2.38692124945801935214171434196, 2.44347211532622590679067670518, 2.59083109891444334535273636167, 2.75333119558039016847702495297, 3.08634921290407597368939515914, 3.10695673882521244508692296655, 3.11502646547986903447376276696, 3.32265376748223940441548832178, 3.42446971051215239225600529119, 3.60481549083769462291188564087, 3.81540857071461933515989233314, 4.11170392903757222581133334658

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

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