Properties

Label 20-138e10-1.1-c1e10-0-1
Degree $20$
Conductor $2.505\times 10^{21}$
Sign $1$
Analytic cond. $2.63974$
Root an. cond. $1.04973$
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  − 2-s + 3-s − 2·5-s − 6-s + 2·10-s + 11·11-s + 13·13-s − 2·15-s − 24·17-s − 14·19-s − 11·22-s − 10·23-s − 17·25-s − 13·26-s + 13·29-s + 2·30-s + 8·31-s + 11·33-s + 24·34-s − 13·37-s + 14·38-s + 13·39-s − 10·41-s − 8·43-s + 10·46-s − 8·47-s + 18·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.894·5-s − 0.408·6-s + 0.632·10-s + 3.31·11-s + 3.60·13-s − 0.516·15-s − 5.82·17-s − 3.21·19-s − 2.34·22-s − 2.08·23-s − 3.39·25-s − 2.54·26-s + 2.41·29-s + 0.365·30-s + 1.43·31-s + 1.91·33-s + 4.11·34-s − 2.13·37-s + 2.27·38-s + 2.08·39-s − 1.56·41-s − 1.21·43-s + 1.47·46-s − 1.16·47-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(2.63974\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{10} \cdot 3^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3705594725\)
\(L(\frac12)\) \(\approx\) \(0.3705594725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
23 \( 1 + 10 T + 34 T^{2} + 131 T^{3} + 1442 T^{4} + 9767 T^{5} + 1442 p T^{6} + 131 p^{2} T^{7} + 34 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 + 2 T + 21 T^{2} + 32 T^{3} + 234 T^{4} + 264 T^{5} + 1756 T^{6} + 1422 T^{7} + 10432 T^{8} + 6384 T^{9} + 54361 T^{10} + 6384 p T^{11} + 10432 p^{2} T^{12} + 1422 p^{3} T^{13} + 1756 p^{4} T^{14} + 264 p^{5} T^{15} + 234 p^{6} T^{16} + 32 p^{7} T^{17} + 21 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
7 \( 1 - 18 T^{2} - 22 T^{3} + 170 T^{4} + 176 T^{5} - 1201 T^{6} - 66 T^{7} + 13808 T^{8} - 4378 T^{9} - 119503 T^{10} - 4378 p T^{11} + 13808 p^{2} T^{12} - 66 p^{3} T^{13} - 1201 p^{4} T^{14} + 176 p^{5} T^{15} + 170 p^{6} T^{16} - 22 p^{7} T^{17} - 18 p^{8} T^{18} + p^{10} T^{20} \)
11 \( 1 - p T + 4 p T^{2} - 4 p T^{3} - 36 p T^{4} + 218 p T^{5} - 50 p^{2} T^{6} + 28 p^{2} T^{7} + 259 p^{2} T^{8} - 939 p^{2} T^{9} + 2124 p^{2} T^{10} - 939 p^{3} T^{11} + 259 p^{4} T^{12} + 28 p^{5} T^{13} - 50 p^{6} T^{14} + 218 p^{6} T^{15} - 36 p^{7} T^{16} - 4 p^{8} T^{17} + 4 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \)
13 \( 1 - p T + 68 T^{2} - 121 T^{3} - 873 T^{4} + 8390 T^{5} - 34053 T^{6} + 62579 T^{7} + 131840 T^{8} - 1594427 T^{9} + 7294143 T^{10} - 1594427 p T^{11} + 131840 p^{2} T^{12} + 62579 p^{3} T^{13} - 34053 p^{4} T^{14} + 8390 p^{5} T^{15} - 873 p^{6} T^{16} - 121 p^{7} T^{17} + 68 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \)
17 \( 1 + 24 T + 262 T^{2} + 1766 T^{3} + 8263 T^{4} + 25873 T^{5} + 27963 T^{6} - 254049 T^{7} - 2176089 T^{8} - 11542392 T^{9} - 51031531 T^{10} - 11542392 p T^{11} - 2176089 p^{2} T^{12} - 254049 p^{3} T^{13} + 27963 p^{4} T^{14} + 25873 p^{5} T^{15} + 8263 p^{6} T^{16} + 1766 p^{7} T^{17} + 262 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 14 T + 45 T^{2} - 252 T^{3} - 1776 T^{4} + 1462 T^{5} + 36590 T^{6} + 45076 T^{7} - 361828 T^{8} + 215942 T^{9} + 9800889 T^{10} + 215942 p T^{11} - 361828 p^{2} T^{12} + 45076 p^{3} T^{13} + 36590 p^{4} T^{14} + 1462 p^{5} T^{15} - 1776 p^{6} T^{16} - 252 p^{7} T^{17} + 45 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 13 T + 41 T^{2} + 240 T^{3} - 1405 T^{4} - 9419 T^{5} + 62410 T^{6} + 364096 T^{7} - 3635431 T^{8} - 1467312 T^{9} + 99241429 T^{10} - 1467312 p T^{11} - 3635431 p^{2} T^{12} + 364096 p^{3} T^{13} + 62410 p^{4} T^{14} - 9419 p^{5} T^{15} - 1405 p^{6} T^{16} + 240 p^{7} T^{17} + 41 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 8 T - 77 T^{2} + 974 T^{3} + 1492 T^{4} - 56584 T^{5} + 107594 T^{6} + 1874024 T^{7} - 9959136 T^{8} - 25696906 T^{9} + 410937273 T^{10} - 25696906 p T^{11} - 9959136 p^{2} T^{12} + 1874024 p^{3} T^{13} + 107594 p^{4} T^{14} - 56584 p^{5} T^{15} + 1492 p^{6} T^{16} + 974 p^{7} T^{17} - 77 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 13 T + 33 T^{2} - 657 T^{3} - 4130 T^{4} + 16390 T^{5} + 255924 T^{6} + 349433 T^{7} - 4984034 T^{8} - 10629229 T^{9} + 104038681 T^{10} - 10629229 p T^{11} - 4984034 p^{2} T^{12} + 349433 p^{3} T^{13} + 255924 p^{4} T^{14} + 16390 p^{5} T^{15} - 4130 p^{6} T^{16} - 657 p^{7} T^{17} + 33 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 10 T + 136 T^{2} + 1643 T^{3} + 16123 T^{4} + 150363 T^{5} + 1259619 T^{6} + 10164498 T^{7} + 75970831 T^{8} + 528150586 T^{9} + 3482845541 T^{10} + 528150586 p T^{11} + 75970831 p^{2} T^{12} + 10164498 p^{3} T^{13} + 1259619 p^{4} T^{14} + 150363 p^{5} T^{15} + 16123 p^{6} T^{16} + 1643 p^{7} T^{17} + 136 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 8 T - 12 T^{2} - 561 T^{3} - 3609 T^{4} - 4485 T^{5} + 139195 T^{6} + 1135662 T^{7} + 3523939 T^{8} - 23870432 T^{9} - 324087611 T^{10} - 23870432 p T^{11} + 3523939 p^{2} T^{12} + 1135662 p^{3} T^{13} + 139195 p^{4} T^{14} - 4485 p^{5} T^{15} - 3609 p^{6} T^{16} - 561 p^{7} T^{17} - 12 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
47 \( ( 1 + 4 T + 138 T^{2} + 670 T^{3} + 11095 T^{4} + 40491 T^{5} + 11095 p T^{6} + 670 p^{2} T^{7} + 138 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
53 \( 1 + 35 T + 534 T^{2} + 4691 T^{3} + 30063 T^{4} + 190332 T^{5} + 837769 T^{6} - 5213181 T^{7} - 124869332 T^{8} - 1080670107 T^{9} - 7457270765 T^{10} - 1080670107 p T^{11} - 124869332 p^{2} T^{12} - 5213181 p^{3} T^{13} + 837769 p^{4} T^{14} + 190332 p^{5} T^{15} + 30063 p^{6} T^{16} + 4691 p^{7} T^{17} + 534 p^{8} T^{18} + 35 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 37 T + 650 T^{2} - 6918 T^{3} + 45103 T^{4} - 118937 T^{5} - 790848 T^{6} + 6814576 T^{7} + 912696 p T^{8} - 1457897138 T^{9} + 14750071633 T^{10} - 1457897138 p T^{11} + 912696 p^{3} T^{12} + 6814576 p^{3} T^{13} - 790848 p^{4} T^{14} - 118937 p^{5} T^{15} + 45103 p^{6} T^{16} - 6918 p^{7} T^{17} + 650 p^{8} T^{18} - 37 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 2 T + 86 T^{2} + 50 T^{3} + 4908 T^{4} - 31228 T^{5} + 212741 T^{6} - 1310478 T^{7} + 12234758 T^{8} - 31646784 T^{9} + 1455356297 T^{10} - 31646784 p T^{11} + 12234758 p^{2} T^{12} - 1310478 p^{3} T^{13} + 212741 p^{4} T^{14} - 31228 p^{5} T^{15} + 4908 p^{6} T^{16} + 50 p^{7} T^{17} + 86 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 14 T + 195 T^{2} - 3354 T^{3} + 39633 T^{4} - 419277 T^{5} + 4814065 T^{6} - 47450510 T^{7} + 426590845 T^{8} - 3846869807 T^{9} + 33199889581 T^{10} - 3846869807 p T^{11} + 426590845 p^{2} T^{12} - 47450510 p^{3} T^{13} + 4814065 p^{4} T^{14} - 419277 p^{5} T^{15} + 39633 p^{6} T^{16} - 3354 p^{7} T^{17} + 195 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 44 T + 897 T^{2} - 12210 T^{3} + 143289 T^{4} - 1622126 T^{5} + 16400193 T^{6} - 139879762 T^{7} + 1112568279 T^{8} - 9545296970 T^{9} + 83433721007 T^{10} - 9545296970 p T^{11} + 1112568279 p^{2} T^{12} - 139879762 p^{3} T^{13} + 16400193 p^{4} T^{14} - 1622126 p^{5} T^{15} + 143289 p^{6} T^{16} - 12210 p^{7} T^{17} + 897 p^{8} T^{18} - 44 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 + 49 T + 1272 T^{2} + 23111 T^{3} + 334395 T^{4} + 4161110 T^{5} + 46769369 T^{6} + 488164589 T^{7} + 4788201674 T^{8} + 44338020863 T^{9} + 388881210783 T^{10} + 44338020863 p T^{11} + 4788201674 p^{2} T^{12} + 488164589 p^{3} T^{13} + 46769369 p^{4} T^{14} + 4161110 p^{5} T^{15} + 334395 p^{6} T^{16} + 23111 p^{7} T^{17} + 1272 p^{8} T^{18} + 49 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 8 T + 227 T^{2} + 282 T^{3} + 24902 T^{4} - 31358 T^{5} + 2808680 T^{6} - 9747498 T^{7} + 221285300 T^{8} - 1405676580 T^{9} + 17486359837 T^{10} - 1405676580 p T^{11} + 221285300 p^{2} T^{12} - 9747498 p^{3} T^{13} + 2808680 p^{4} T^{14} - 31358 p^{5} T^{15} + 24902 p^{6} T^{16} + 282 p^{7} T^{17} + 227 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 + 17 T + 52 T^{2} - 406 T^{3} + 244 T^{4} + 1029 T^{5} - 761770 T^{6} - 8004680 T^{7} + 16074276 T^{8} + 470096066 T^{9} + 2367060235 T^{10} + 470096066 p T^{11} + 16074276 p^{2} T^{12} - 8004680 p^{3} T^{13} - 761770 p^{4} T^{14} + 1029 p^{5} T^{15} + 244 p^{6} T^{16} - 406 p^{7} T^{17} + 52 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 59 T + 1742 T^{2} - 36334 T^{3} + 614746 T^{4} - 8918095 T^{5} + 114846706 T^{6} - 1347527760 T^{7} + 14656918836 T^{8} - 150039947354 T^{9} + 1454906272975 T^{10} - 150039947354 p T^{11} + 14656918836 p^{2} T^{12} - 1347527760 p^{3} T^{13} + 114846706 p^{4} T^{14} - 8918095 p^{5} T^{15} + 614746 p^{6} T^{16} - 36334 p^{7} T^{17} + 1742 p^{8} T^{18} - 59 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 21 T + 300 T^{2} + 5341 T^{3} + 48477 T^{4} + 384748 T^{5} + 3678795 T^{6} - 8287925 T^{7} - 183989272 T^{8} - 2558529435 T^{9} - 54591704045 T^{10} - 2558529435 p T^{11} - 183989272 p^{2} T^{12} - 8287925 p^{3} T^{13} + 3678795 p^{4} T^{14} + 384748 p^{5} T^{15} + 48477 p^{6} T^{16} + 5341 p^{7} T^{17} + 300 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.29038655111059805249455139383, −5.00404828242109854314901002784, −4.95529860332498646354984784881, −4.76973497030798439490305308654, −4.69564151398396659296988759820, −4.55646159590577678250025061701, −4.22967354087050032802936743124, −4.21918002703971402997536291996, −4.10396741561284494546443903544, −3.95403265836380024578165053345, −3.90856011211871057868797474926, −3.76564756702632490519309041234, −3.71165325706880468394763523266, −3.65212002451216462848031331302, −3.42794836077806491681746206815, −2.95948536190267993125782890777, −2.76247369260552375834314944951, −2.59219720375274437005176621041, −2.29985399925755402952844568343, −2.06107111598418183696851393998, −1.93421120601844444960567385219, −1.80815365556608671869491429685, −1.60348294142352999779041223287, −1.35833613407152544146735100438, −0.35217438654691582087542550619, 0.35217438654691582087542550619, 1.35833613407152544146735100438, 1.60348294142352999779041223287, 1.80815365556608671869491429685, 1.93421120601844444960567385219, 2.06107111598418183696851393998, 2.29985399925755402952844568343, 2.59219720375274437005176621041, 2.76247369260552375834314944951, 2.95948536190267993125782890777, 3.42794836077806491681746206815, 3.65212002451216462848031331302, 3.71165325706880468394763523266, 3.76564756702632490519309041234, 3.90856011211871057868797474926, 3.95403265836380024578165053345, 4.10396741561284494546443903544, 4.21918002703971402997536291996, 4.22967354087050032802936743124, 4.55646159590577678250025061701, 4.69564151398396659296988759820, 4.76973497030798439490305308654, 4.95529860332498646354984784881, 5.00404828242109854314901002784, 5.29038655111059805249455139383

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

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