Properties

Degree 20
Conductor $ 2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s + 8·3-s + 55·4-s + 10·5-s − 80·6-s + 3·7-s − 220·8-s + 24·9-s − 100·10-s + 10·11-s + 440·12-s + 7·13-s − 30·14-s + 80·15-s + 715·16-s + 2·17-s − 240·18-s − 7·19-s + 550·20-s + 24·21-s − 100·22-s + 12·23-s − 1.76e3·24-s + 55·25-s − 70·26-s + 21·27-s + 165·28-s + ⋯
L(s)  = 1  − 7.07·2-s + 4.61·3-s + 55/2·4-s + 4.47·5-s − 32.6·6-s + 1.13·7-s − 77.7·8-s + 8·9-s − 31.6·10-s + 3.01·11-s + 127.·12-s + 1.94·13-s − 8.01·14-s + 20.6·15-s + 178.·16-s + 0.485·17-s − 56.5·18-s − 1.60·19-s + 122.·20-s + 5.23·21-s − 21.3·22-s + 2.50·23-s − 359.·24-s + 11·25-s − 13.7·26-s + 4.04·27-s + 31.1·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4730} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $\approx$  $170.7851625$
$L(\frac12)$  $\approx$  $170.7851625$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\(F_p(T)\) is a polynomial of degree 20. If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{10} \)
5 \( ( 1 - T )^{10} \)
11 \( ( 1 - T )^{10} \)
43 \( ( 1 - T )^{10} \)
good3 \( 1 - 8 T + 40 T^{2} - 149 T^{3} + 155 p T^{4} - 1271 T^{5} + 349 p^{2} T^{6} - 263 p^{3} T^{7} + 14782 T^{8} - 352 p^{4} T^{9} + 1894 p^{3} T^{10} - 352 p^{5} T^{11} + 14782 p^{2} T^{12} - 263 p^{6} T^{13} + 349 p^{6} T^{14} - 1271 p^{5} T^{15} + 155 p^{7} T^{16} - 149 p^{7} T^{17} + 40 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
7 \( 1 - 3 T + 4 p T^{2} - 64 T^{3} + 366 T^{4} - 501 T^{5} + 2879 T^{6} - 1424 T^{7} + 17125 T^{8} + 5441 T^{9} + 107983 T^{10} + 5441 p T^{11} + 17125 p^{2} T^{12} - 1424 p^{3} T^{13} + 2879 p^{4} T^{14} - 501 p^{5} T^{15} + 366 p^{6} T^{16} - 64 p^{7} T^{17} + 4 p^{9} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 - 7 T + 68 T^{2} - 477 T^{3} + 2739 T^{4} - 15750 T^{5} + 76220 T^{6} - 354402 T^{7} + 1523764 T^{8} - 5999736 T^{9} + 22762496 T^{10} - 5999736 p T^{11} + 1523764 p^{2} T^{12} - 354402 p^{3} T^{13} + 76220 p^{4} T^{14} - 15750 p^{5} T^{15} + 2739 p^{6} T^{16} - 477 p^{7} T^{17} + 68 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 2 T + 96 T^{2} - 145 T^{3} + 4785 T^{4} - 5793 T^{5} + 161427 T^{6} - 162113 T^{7} + 4038980 T^{8} - 3497112 T^{9} + 77878470 T^{10} - 3497112 p T^{11} + 4038980 p^{2} T^{12} - 162113 p^{3} T^{13} + 161427 p^{4} T^{14} - 5793 p^{5} T^{15} + 4785 p^{6} T^{16} - 145 p^{7} T^{17} + 96 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 7 T + 122 T^{2} + 690 T^{3} + 6892 T^{4} + 31513 T^{5} + 238673 T^{6} + 908590 T^{7} + 5931789 T^{8} + 19825389 T^{9} + 120772887 T^{10} + 19825389 p T^{11} + 5931789 p^{2} T^{12} + 908590 p^{3} T^{13} + 238673 p^{4} T^{14} + 31513 p^{5} T^{15} + 6892 p^{6} T^{16} + 690 p^{7} T^{17} + 122 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 - 12 T + 150 T^{2} - 50 p T^{3} + 8249 T^{4} - 41650 T^{5} + 187704 T^{6} - 19830 p T^{7} + 217430 T^{8} + 10641030 T^{9} - 59134492 T^{10} + 10641030 p T^{11} + 217430 p^{2} T^{12} - 19830 p^{4} T^{13} + 187704 p^{4} T^{14} - 41650 p^{5} T^{15} + 8249 p^{6} T^{16} - 50 p^{8} T^{17} + 150 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 + 12 T + 172 T^{2} + 1212 T^{3} + 11748 T^{4} + 71742 T^{5} + 623346 T^{6} + 3458610 T^{7} + 25499875 T^{8} + 122678304 T^{9} + 805916100 T^{10} + 122678304 p T^{11} + 25499875 p^{2} T^{12} + 3458610 p^{3} T^{13} + 623346 p^{4} T^{14} + 71742 p^{5} T^{15} + 11748 p^{6} T^{16} + 1212 p^{7} T^{17} + 172 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 16 T + 268 T^{2} - 92 p T^{3} + 28846 T^{4} - 235464 T^{5} + 1821050 T^{6} - 12362000 T^{7} + 80409761 T^{8} - 479580508 T^{9} + 2766384724 T^{10} - 479580508 p T^{11} + 80409761 p^{2} T^{12} - 12362000 p^{3} T^{13} + 1821050 p^{4} T^{14} - 235464 p^{5} T^{15} + 28846 p^{6} T^{16} - 92 p^{8} T^{17} + 268 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 19 T + 331 T^{2} - 3864 T^{3} + 42257 T^{4} - 382535 T^{5} + 3315536 T^{6} - 25355009 T^{7} + 186527846 T^{8} - 1237987393 T^{9} + 7903177706 T^{10} - 1237987393 p T^{11} + 186527846 p^{2} T^{12} - 25355009 p^{3} T^{13} + 3315536 p^{4} T^{14} - 382535 p^{5} T^{15} + 42257 p^{6} T^{16} - 3864 p^{7} T^{17} + 331 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 - 9 T + 202 T^{2} - 43 p T^{3} + 22787 T^{4} - 183686 T^{5} + 1804740 T^{6} - 12945466 T^{7} + 107949340 T^{8} - 683450588 T^{9} + 5008974308 T^{10} - 683450588 p T^{11} + 107949340 p^{2} T^{12} - 12945466 p^{3} T^{13} + 1804740 p^{4} T^{14} - 183686 p^{5} T^{15} + 22787 p^{6} T^{16} - 43 p^{8} T^{17} + 202 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - 29 T + 630 T^{2} - 10178 T^{3} + 139651 T^{4} - 34776 p T^{5} + 17082417 T^{6} - 159419870 T^{7} + 1358654480 T^{8} - 10549289179 T^{9} + 75551985562 T^{10} - 10549289179 p T^{11} + 1358654480 p^{2} T^{12} - 159419870 p^{3} T^{13} + 17082417 p^{4} T^{14} - 34776 p^{6} T^{15} + 139651 p^{6} T^{16} - 10178 p^{7} T^{17} + 630 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 6 T + 228 T^{2} - 823 T^{3} + 26913 T^{4} - 84095 T^{5} + 2484181 T^{6} - 7507459 T^{7} + 178011076 T^{8} - 475542760 T^{9} + 10211795688 T^{10} - 475542760 p T^{11} + 178011076 p^{2} T^{12} - 7507459 p^{3} T^{13} + 2484181 p^{4} T^{14} - 84095 p^{5} T^{15} + 26913 p^{6} T^{16} - 823 p^{7} T^{17} + 228 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 29 T + 672 T^{2} - 11632 T^{3} + 174877 T^{4} - 2245112 T^{5} + 26055127 T^{6} - 270374524 T^{7} + 2575298424 T^{8} - 22349080157 T^{9} + 179470459326 T^{10} - 22349080157 p T^{11} + 2575298424 p^{2} T^{12} - 270374524 p^{3} T^{13} + 26055127 p^{4} T^{14} - 2245112 p^{5} T^{15} + 174877 p^{6} T^{16} - 11632 p^{7} T^{17} + 672 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 4 T + 388 T^{2} + 868 T^{3} + 70996 T^{4} + 57814 T^{5} + 8392678 T^{6} - 1183390 T^{7} + 734050851 T^{8} - 410475032 T^{9} + 50269115596 T^{10} - 410475032 p T^{11} + 734050851 p^{2} T^{12} - 1183390 p^{3} T^{13} + 8392678 p^{4} T^{14} + 57814 p^{5} T^{15} + 70996 p^{6} T^{16} + 868 p^{7} T^{17} + 388 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 45 T + 1362 T^{2} - 29555 T^{3} + 529465 T^{4} - 7948376 T^{5} + 104849664 T^{6} - 1219722800 T^{7} + 12798796014 T^{8} - 120871853754 T^{9} + 1040010415772 T^{10} - 120871853754 p T^{11} + 12798796014 p^{2} T^{12} - 1219722800 p^{3} T^{13} + 104849664 p^{4} T^{14} - 7948376 p^{5} T^{15} + 529465 p^{6} T^{16} - 29555 p^{7} T^{17} + 1362 p^{8} T^{18} - 45 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 18 T + 336 T^{2} + 4251 T^{3} + 58715 T^{4} + 643707 T^{5} + 7369235 T^{6} + 72030587 T^{7} + 721946446 T^{8} + 6325165830 T^{9} + 56555308584 T^{10} + 6325165830 p T^{11} + 721946446 p^{2} T^{12} + 72030587 p^{3} T^{13} + 7369235 p^{4} T^{14} + 643707 p^{5} T^{15} + 58715 p^{6} T^{16} + 4251 p^{7} T^{17} + 336 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 3 T + 418 T^{2} - 727 T^{3} + 80673 T^{4} - 37760 T^{5} + 9681104 T^{6} + 7643944 T^{7} + 853999630 T^{8} + 1455931258 T^{9} + 64655784028 T^{10} + 1455931258 p T^{11} + 853999630 p^{2} T^{12} + 7643944 p^{3} T^{13} + 9681104 p^{4} T^{14} - 37760 p^{5} T^{15} + 80673 p^{6} T^{16} - 727 p^{7} T^{17} + 418 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 14 T + 337 T^{2} + 4599 T^{3} + 55352 T^{4} + 681756 T^{5} + 6101048 T^{6} + 61563193 T^{7} + 541631883 T^{8} + 4355926086 T^{9} + 43938008184 T^{10} + 4355926086 p T^{11} + 541631883 p^{2} T^{12} + 61563193 p^{3} T^{13} + 6101048 p^{4} T^{14} + 681756 p^{5} T^{15} + 55352 p^{6} T^{16} + 4599 p^{7} T^{17} + 337 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 23 T + 776 T^{2} - 13490 T^{3} + 267861 T^{4} - 3724064 T^{5} + 55138019 T^{6} - 636062758 T^{7} + 7617153620 T^{8} - 74270783979 T^{9} + 745297454950 T^{10} - 74270783979 p T^{11} + 7617153620 p^{2} T^{12} - 636062758 p^{3} T^{13} + 55138019 p^{4} T^{14} - 3724064 p^{5} T^{15} + 267861 p^{6} T^{16} - 13490 p^{7} T^{17} + 776 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - T + 443 T^{2} - 382 T^{3} + 99407 T^{4} - 96203 T^{5} + 15048020 T^{6} - 17286665 T^{7} + 1744367712 T^{8} - 2164824613 T^{9} + 167746942306 T^{10} - 2164824613 p T^{11} + 1744367712 p^{2} T^{12} - 17286665 p^{3} T^{13} + 15048020 p^{4} T^{14} - 96203 p^{5} T^{15} + 99407 p^{6} T^{16} - 382 p^{7} T^{17} + 443 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 30 T + 1006 T^{2} - 20394 T^{3} + 411581 T^{4} - 6457164 T^{5} + 98205176 T^{6} - 1265341780 T^{7} + 15688018626 T^{8} - 170791635800 T^{9} + 1785886492596 T^{10} - 170791635800 p T^{11} + 15688018626 p^{2} T^{12} - 1265341780 p^{3} T^{13} + 98205176 p^{4} T^{14} - 6457164 p^{5} T^{15} + 411581 p^{6} T^{16} - 20394 p^{7} T^{17} + 1006 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.69174817189965820865802188913, −2.53958591765431849897835879777, −2.52277098840153535665104418258, −2.51197565517043102503328696194, −2.51003255113544496841817516575, −2.22614517390393884850872704047, −2.05690367716154659941945627559, −1.99935709316555282938812554830, −1.96130245490876590485040592532, −1.93731653990956197826984733401, −1.88947716941194334933834353847, −1.88917908675220828248916331041, −1.68016166398938486439477754057, −1.64113162598623176045817451033, −1.52640497433050082709337696877, −1.20781715940695874317394570455, −1.18589860329243067506047997612, −0.975291240593934978847746102773, −0.958065939484708868057225303540, −0.920448080193192325138220393818, −0.845090759264541758927565292446, −0.811668808656325173656208869953, −0.66031632370940445215224059993, −0.47341552674166497677161911719, −0.44519608781075129718803084684, 0.44519608781075129718803084684, 0.47341552674166497677161911719, 0.66031632370940445215224059993, 0.811668808656325173656208869953, 0.845090759264541758927565292446, 0.920448080193192325138220393818, 0.958065939484708868057225303540, 0.975291240593934978847746102773, 1.18589860329243067506047997612, 1.20781715940695874317394570455, 1.52640497433050082709337696877, 1.64113162598623176045817451033, 1.68016166398938486439477754057, 1.88917908675220828248916331041, 1.88947716941194334933834353847, 1.93731653990956197826984733401, 1.96130245490876590485040592532, 1.99935709316555282938812554830, 2.05690367716154659941945627559, 2.22614517390393884850872704047, 2.51003255113544496841817516575, 2.51197565517043102503328696194, 2.52277098840153535665104418258, 2.53958591765431849897835879777, 2.69174817189965820865802188913

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

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