Properties

Label 16-4026e8-1.1-c1e8-0-0
Degree $16$
Conductor $6.902\times 10^{28}$
Sign $1$
Analytic cond. $1.14079\times 10^{12}$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 8·3-s + 36·4-s + 5·5-s + 64·6-s + 13·7-s + 120·8-s + 36·9-s + 40·10-s − 8·11-s + 288·12-s + 10·13-s + 104·14-s + 40·15-s + 330·16-s + 4·17-s + 288·18-s + 11·19-s + 180·20-s + 104·21-s − 64·22-s + 2·23-s + 960·24-s + 4·25-s + 80·26-s + 120·27-s + 468·28-s + ⋯
L(s)  = 1  + 5.65·2-s + 4.61·3-s + 18·4-s + 2.23·5-s + 26.1·6-s + 4.91·7-s + 42.4·8-s + 12·9-s + 12.6·10-s − 2.41·11-s + 83.1·12-s + 2.77·13-s + 27.7·14-s + 10.3·15-s + 82.5·16-s + 0.970·17-s + 67.8·18-s + 2.52·19-s + 40.2·20-s + 22.6·21-s − 13.6·22-s + 0.417·23-s + 195.·24-s + 4/5·25-s + 15.6·26-s + 23.0·27-s + 88.4·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{8} \cdot 61^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{8} \cdot 61^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 11^{8} \cdot 61^{8}\)
Sign: $1$
Analytic conductor: \(1.14079\times 10^{12}\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 11^{8} \cdot 61^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(150217.8495\)
\(L(\frac12)\) \(\approx\) \(150217.8495\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
3 \( ( 1 - T )^{8} \)
11 \( ( 1 + T )^{8} \)
61 \( ( 1 - T )^{8} \)
good5 \( 1 - p T + 21 T^{2} - 61 T^{3} + 192 T^{4} - 506 T^{5} + 267 p T^{6} - 588 p T^{7} + 6862 T^{8} - 588 p^{2} T^{9} + 267 p^{3} T^{10} - 506 p^{3} T^{11} + 192 p^{4} T^{12} - 61 p^{5} T^{13} + 21 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \)
7 \( 1 - 13 T + 104 T^{2} - 611 T^{3} + 2950 T^{4} - 12084 T^{5} + 43244 T^{6} - 136182 T^{7} + 381994 T^{8} - 136182 p T^{9} + 43244 p^{2} T^{10} - 12084 p^{3} T^{11} + 2950 p^{4} T^{12} - 611 p^{5} T^{13} + 104 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 10 T + 107 T^{2} - 712 T^{3} + 4503 T^{4} - 22801 T^{5} + 107360 T^{6} - 441362 T^{7} + 1682040 T^{8} - 441362 p T^{9} + 107360 p^{2} T^{10} - 22801 p^{3} T^{11} + 4503 p^{4} T^{12} - 712 p^{5} T^{13} + 107 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 4 T + 73 T^{2} - 275 T^{3} + 3198 T^{4} - 10516 T^{5} + 89009 T^{6} - 258321 T^{7} + 1803046 T^{8} - 258321 p T^{9} + 89009 p^{2} T^{10} - 10516 p^{3} T^{11} + 3198 p^{4} T^{12} - 275 p^{5} T^{13} + 73 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 11 T + 118 T^{2} - 795 T^{3} + 5394 T^{4} - 28896 T^{5} + 8230 p T^{6} - 714344 T^{7} + 3336682 T^{8} - 714344 p T^{9} + 8230 p^{3} T^{10} - 28896 p^{3} T^{11} + 5394 p^{4} T^{12} - 795 p^{5} T^{13} + 118 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 2 T + 135 T^{2} - 249 T^{3} + 8858 T^{4} - 14627 T^{5} + 361913 T^{6} - 518212 T^{7} + 10011114 T^{8} - 518212 p T^{9} + 361913 p^{2} T^{10} - 14627 p^{3} T^{11} + 8858 p^{4} T^{12} - 249 p^{5} T^{13} + 135 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 10 T + 154 T^{2} - 1061 T^{3} + 10523 T^{4} - 62366 T^{5} + 493342 T^{6} - 2543295 T^{7} + 16694912 T^{8} - 2543295 p T^{9} + 493342 p^{2} T^{10} - 62366 p^{3} T^{11} + 10523 p^{4} T^{12} - 1061 p^{5} T^{13} + 154 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 9 T + 197 T^{2} - 1601 T^{3} + 594 p T^{4} - 130354 T^{5} + 1050057 T^{6} - 6298538 T^{7} + 39612950 T^{8} - 6298538 p T^{9} + 1050057 p^{2} T^{10} - 130354 p^{3} T^{11} + 594 p^{5} T^{12} - 1601 p^{5} T^{13} + 197 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 9 T + 98 T^{2} - 635 T^{3} + 5646 T^{4} - 26596 T^{5} + 181832 T^{6} - 675000 T^{7} + 6018774 T^{8} - 675000 p T^{9} + 181832 p^{2} T^{10} - 26596 p^{3} T^{11} + 5646 p^{4} T^{12} - 635 p^{5} T^{13} + 98 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 3 T + 262 T^{2} - 677 T^{3} + 32119 T^{4} - 71264 T^{5} + 2398904 T^{6} - 4507824 T^{7} + 119184500 T^{8} - 4507824 p T^{9} + 2398904 p^{2} T^{10} - 71264 p^{3} T^{11} + 32119 p^{4} T^{12} - 677 p^{5} T^{13} + 262 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 16 T + 205 T^{2} - 1085 T^{3} + 3792 T^{4} + 28237 T^{5} - 134453 T^{6} + 817270 T^{7} + 4143710 T^{8} + 817270 p T^{9} - 134453 p^{2} T^{10} + 28237 p^{3} T^{11} + 3792 p^{4} T^{12} - 1085 p^{5} T^{13} + 205 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 16 T + 206 T^{2} + 1802 T^{3} + 12129 T^{4} + 38819 T^{5} - 131698 T^{6} - 3691287 T^{7} - 32095452 T^{8} - 3691287 p T^{9} - 131698 p^{2} T^{10} + 38819 p^{3} T^{11} + 12129 p^{4} T^{12} + 1802 p^{5} T^{13} + 206 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 7 T + 208 T^{2} - 1453 T^{3} + 24474 T^{4} - 146432 T^{5} + 1962770 T^{6} - 10336604 T^{7} + 117577006 T^{8} - 10336604 p T^{9} + 1962770 p^{2} T^{10} - 146432 p^{3} T^{11} + 24474 p^{4} T^{12} - 1453 p^{5} T^{13} + 208 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 14 T + 489 T^{2} + 5476 T^{3} + 102809 T^{4} + 934581 T^{5} + 12232504 T^{6} + 1529428 p T^{7} + 904300590 T^{8} + 1529428 p^{2} T^{9} + 12232504 p^{2} T^{10} + 934581 p^{3} T^{11} + 102809 p^{4} T^{12} + 5476 p^{5} T^{13} + 489 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 8 T + 290 T^{2} - 2224 T^{3} + 49393 T^{4} - 334771 T^{5} + 5412530 T^{6} - 32410203 T^{7} + 430011908 T^{8} - 32410203 p T^{9} + 5412530 p^{2} T^{10} - 334771 p^{3} T^{11} + 49393 p^{4} T^{12} - 2224 p^{5} T^{13} + 290 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 11 T + 302 T^{2} - 3528 T^{3} + 53098 T^{4} - 539286 T^{5} + 6310066 T^{6} - 54605825 T^{7} + 526561322 T^{8} - 54605825 p T^{9} + 6310066 p^{2} T^{10} - 539286 p^{3} T^{11} + 53098 p^{4} T^{12} - 3528 p^{5} T^{13} + 302 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 14 T + 343 T^{2} - 3319 T^{3} + 45652 T^{4} - 293923 T^{5} + 3108441 T^{6} - 13508928 T^{7} + 176953174 T^{8} - 13508928 p T^{9} + 3108441 p^{2} T^{10} - 293923 p^{3} T^{11} + 45652 p^{4} T^{12} - 3319 p^{5} T^{13} + 343 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 22 T + 620 T^{2} - 10382 T^{3} + 2169 p T^{4} - 2215601 T^{5} + 27106408 T^{6} - 277046829 T^{7} + 2677646568 T^{8} - 277046829 p T^{9} + 27106408 p^{2} T^{10} - 2215601 p^{3} T^{11} + 2169 p^{5} T^{12} - 10382 p^{5} T^{13} + 620 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 16 T + 285 T^{2} + 3041 T^{3} + 38374 T^{4} + 345939 T^{5} + 4548555 T^{6} + 42313374 T^{7} + 456851282 T^{8} + 42313374 p T^{9} + 4548555 p^{2} T^{10} + 345939 p^{3} T^{11} + 38374 p^{4} T^{12} + 3041 p^{5} T^{13} + 285 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - T + 221 T^{2} + 1243 T^{3} + 14742 T^{4} + 211258 T^{5} + 1643035 T^{6} + 2642040 T^{7} + 244871018 T^{8} + 2642040 p T^{9} + 1643035 p^{2} T^{10} + 211258 p^{3} T^{11} + 14742 p^{4} T^{12} + 1243 p^{5} T^{13} + 221 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 24 T + 345 T^{2} - 4508 T^{3} + 52531 T^{4} - 526413 T^{5} + 5608724 T^{6} - 52037408 T^{7} + 460813184 T^{8} - 52037408 p T^{9} + 5608724 p^{2} T^{10} - 526413 p^{3} T^{11} + 52531 p^{4} T^{12} - 4508 p^{5} T^{13} + 345 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.56779305901228411941021536853, −3.17628595845764438463547634927, −3.14984265359743266265843895483, −3.11944747111452513126741662238, −3.05374266149935148836287313963, −3.04061676266762195898243303913, −2.90522371859641483678065134706, −2.79312609424135663538479251537, −2.74795188337003411605783503529, −2.39029975567659190011560029550, −2.26971931017383605098158062265, −2.18344814520338804248528388513, −2.16600607829489477691530340275, −2.13468475178249488809894661431, −2.06456731944300826561798943850, −2.04997348249997116548951699412, −1.96745843863188391769848713731, −1.39561615029384657038657488329, −1.36273614547147086299217959479, −1.34070075489715150602486403815, −1.29647763776162242271983671926, −1.28528413614215449769657897313, −1.03747505625004592100475165422, −0.921836606494412742066908515995, −0.836197649367144851647975569010, 0.836197649367144851647975569010, 0.921836606494412742066908515995, 1.03747505625004592100475165422, 1.28528413614215449769657897313, 1.29647763776162242271983671926, 1.34070075489715150602486403815, 1.36273614547147086299217959479, 1.39561615029384657038657488329, 1.96745843863188391769848713731, 2.04997348249997116548951699412, 2.06456731944300826561798943850, 2.13468475178249488809894661431, 2.16600607829489477691530340275, 2.18344814520338804248528388513, 2.26971931017383605098158062265, 2.39029975567659190011560029550, 2.74795188337003411605783503529, 2.79312609424135663538479251537, 2.90522371859641483678065134706, 3.04061676266762195898243303913, 3.05374266149935148836287313963, 3.11944747111452513126741662238, 3.14984265359743266265843895483, 3.17628595845764438463547634927, 3.56779305901228411941021536853

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

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