Properties

Degree 20
Conductor $ 2^{40} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 6·4-s − 2·5-s + 4·6-s − 24·8-s + 2·9-s + 4·10-s + 18·11-s − 12·12-s − 2·13-s + 4·15-s + 116·16-s − 4·17-s − 4·18-s − 26·19-s − 12·20-s − 36·22-s + 48·24-s + 2·25-s + 4·26-s + 42·27-s − 202·29-s − 8·30-s + 368·31-s − 56·32-s − 36·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s + 3/4·4-s − 0.178·5-s + 0.272·6-s − 1.06·8-s + 2/27·9-s + 0.126·10-s + 0.493·11-s − 0.288·12-s − 0.0426·13-s + 0.0688·15-s + 1.81·16-s − 0.0570·17-s − 0.0523·18-s − 0.313·19-s − 0.134·20-s − 0.348·22-s + 0.408·24-s + 0.0159·25-s + 0.0301·26-s + 0.299·27-s − 1.29·29-s − 0.0486·30-s + 2.13·31-s − 0.309·32-s − 0.189·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{40}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{16} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 2^{40} ,\ ( \ : [3/2]^{10} ),\ 1 )$
$L(2)$  $\approx$  $0.637296$
$L(\frac12)$  $\approx$  $0.637296$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 20. If $p = 2$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 \( 1 + p T - p T^{2} + p^{3} T^{3} - 5 p^{3} T^{4} - 11 p^{5} T^{5} - 5 p^{6} T^{6} + p^{9} T^{7} - p^{10} T^{8} + p^{13} T^{9} + p^{15} T^{10} \)
good3 \( 1 + 2 T + 2 T^{2} - 14 p T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 22264 p T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 3046228 p^{3} T^{11} + 57874 p^{6} T^{12} + 22264 p^{10} T^{13} + 3544 p^{12} T^{14} + 760 p^{15} T^{15} - 571 p^{18} T^{16} - 14 p^{22} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20} \)
5 \( 1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 3608 p T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 2301854348 p^{3} T^{11} - 28535006 p^{6} T^{12} - 11981288 p^{9} T^{13} + 530104 p^{12} T^{14} + 3608 p^{16} T^{15} - 13723 p^{18} T^{16} - 966 p^{21} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20} \)
7 \( 1 - 1762 T^{2} + 219995 p T^{4} - 133155368 p T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 440869947922 p^{6} T^{12} - 133155368 p^{13} T^{14} + 219995 p^{19} T^{16} - 1762 p^{24} T^{18} + p^{30} T^{20} \)
11 \( 1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 56924680478 p^{2} T^{8} + 9237744161500 p T^{9} + 18755914132083020 T^{10} + 9237744161500 p^{4} T^{11} - 56924680478 p^{8} T^{12} - 313798751208 p^{9} T^{13} + 5463099864 p^{12} T^{14} + 79597000 p^{15} T^{15} + 4077397 p^{18} T^{16} - 122934 p^{21} T^{17} + 162 p^{24} T^{18} - 18 p^{27} T^{19} + p^{30} T^{20} \)
13 \( 1 + 2 T + 2 T^{2} - 45206 T^{3} - 184727 p T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 29556007880 p T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 1531455561616908 p^{3} T^{11} - 9400034983966 p^{6} T^{12} - 29556007880 p^{10} T^{13} + 1085386040 p^{12} T^{14} + 29395960 p^{15} T^{15} - 184727 p^{19} T^{16} - 45206 p^{21} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20} \)
17 \( ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 98460610 p^{3} T^{6} + 102520 p^{6} T^{7} + 12653 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
19 \( 1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} - 259766590630759364 p^{3} T^{11} + 1011857007777554 p^{6} T^{12} + 36783398837960 p^{9} T^{13} + 143449890200 p^{12} T^{14} + 2461461784 p^{15} T^{15} - 64153371 p^{18} T^{16} - 339906 p^{21} T^{17} + 338 p^{24} T^{18} + 26 p^{27} T^{19} + p^{30} T^{20} \)
23 \( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + 1342371312768300946 p^{6} T^{12} - 73253961622040 p^{12} T^{14} + 2913757597 p^{18} T^{16} - 76386 p^{24} T^{18} + p^{30} T^{20} \)
29 \( 1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} + \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!32\)\( p^{3} T^{11} + 194871598558001506 p^{6} T^{12} + 2529271278095288 p^{9} T^{13} + 40873283338616 p^{12} T^{14} + 239164019416 p^{15} T^{15} + 398569397 p^{18} T^{16} - 1177934 p^{21} T^{17} + 20402 p^{24} T^{18} + 202 p^{27} T^{19} + p^{30} T^{20} \)
31 \( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 7638677322 p^{3} T^{6} - 19809056 p^{6} T^{7} + 134043 p^{9} T^{8} - 184 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
37 \( 1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} + 27985624577691139772 p^{3} T^{11} + 3474549392106364962 p^{6} T^{12} - 5252842710654600 p^{9} T^{13} + 324850634232 p^{12} T^{14} - 153991562664 p^{15} T^{15} + 1630465317 p^{18} T^{16} + 1972962 p^{21} T^{17} + 50 p^{24} T^{18} + 10 p^{27} T^{19} + p^{30} T^{20} \)
41 \( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(14\!\cdots\!14\)\( p^{6} T^{12} - 13934678680622904 p^{12} T^{14} + 97166156061 p^{18} T^{16} - 441018 p^{24} T^{18} + p^{30} T^{20} \)
43 \( 1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} + \)\(17\!\cdots\!68\)\( p^{3} T^{11} + \)\(59\!\cdots\!10\)\( p^{6} T^{12} + 1952595983380873720 p^{9} T^{13} + 6471694737204248 p^{12} T^{14} + 20936462157416 p^{15} T^{15} + 56398378005 p^{18} T^{16} + 132133650 p^{21} T^{17} + 351122 p^{24} T^{18} + 838 p^{27} T^{19} + p^{30} T^{20} \)
47 \( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 90105579914 p^{3} T^{6} + 171516064 p^{6} T^{7} + 462219 p^{9} T^{8} + 472 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
53 \( 1 + 378 T + 71442 T^{2} - 995350 p T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} - \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} - \)\(14\!\cdots\!20\)\( p^{3} T^{11} - 1581505854923305054 p^{6} T^{12} - 1674349213754452168 p^{9} T^{13} + 976891435665272 p^{12} T^{14} - 526686651752 p^{15} T^{15} + 3016286341 p^{18} T^{16} - 995350 p^{22} T^{17} + 71442 p^{24} T^{18} + 378 p^{27} T^{19} + p^{30} T^{20} \)
59 \( 1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} + \)\(37\!\cdots\!12\)\( p T^{9} - \)\(35\!\cdots\!80\)\( p^{2} T^{10} + \)\(37\!\cdots\!12\)\( p^{4} T^{11} + \)\(57\!\cdots\!98\)\( p^{6} T^{12} - 20092577710244830152 p^{9} T^{13} + 78453755015006616 p^{12} T^{14} - 232658117164632 p^{15} T^{15} + 555128806581 p^{18} T^{16} - 989315358 p^{21} T^{17} + 1455218 p^{24} T^{18} - 1706 p^{27} T^{19} + p^{30} T^{20} \)
61 \( 1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} - \)\(32\!\cdots\!00\)\( p^{3} T^{11} + \)\(59\!\cdots\!50\)\( p^{6} T^{12} - 6001380717052735080 p^{9} T^{13} + 58848327585507000 p^{12} T^{14} - 72823214590920 p^{15} T^{15} + 20471098485 p^{18} T^{16} + 45940410 p^{21} T^{17} + 414050 p^{24} T^{18} - 910 p^{27} T^{19} + p^{30} T^{20} \)
67 \( 1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} - \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} - \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} - \)\(17\!\cdots\!64\)\( p^{3} T^{11} + \)\(29\!\cdots\!06\)\( p^{6} T^{12} - \)\(46\!\cdots\!12\)\( p^{9} T^{13} + 724275679990789656 p^{12} T^{14} - 1068751594464168 p^{15} T^{15} + 1338066168261 p^{18} T^{16} - 1530965298 p^{21} T^{17} + 1885682 p^{24} T^{18} - 1942 p^{27} T^{19} + p^{30} T^{20} \)
71 \( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(13\!\cdots\!66\)\( p^{6} T^{12} - 2420243241413642648 p^{12} T^{14} + 3072516920573 p^{18} T^{16} - 2500418 p^{24} T^{18} + p^{30} T^{20} \)
73 \( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(26\!\cdots\!74\)\( p^{6} T^{12} - 4242124717558516472 p^{12} T^{14} + 4627160201821 p^{18} T^{16} - 3134282 p^{24} T^{18} + p^{30} T^{20} \)
79 \( ( 1 + 2208 T + 3816107 T^{2} + 54694528 p T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 4245684014154 p^{3} T^{6} + 54694528 p^{7} T^{7} + 3816107 p^{9} T^{8} + 2208 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
83 \( 1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} + \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} + \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} + \)\(97\!\cdots\!88\)\( p^{3} T^{11} + \)\(89\!\cdots\!74\)\( p^{6} T^{12} + \)\(79\!\cdots\!28\)\( p^{9} T^{13} + 882645219418437336 p^{12} T^{14} + 1191348439341176 p^{15} T^{15} + 1934799974629 p^{18} T^{16} + 2891460918 p^{21} T^{17} + 3281922 p^{24} T^{18} + 2562 p^{27} T^{19} + p^{30} T^{20} \)
89 \( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(69\!\cdots\!98\)\( p^{6} T^{12} - 7720859292932177528 p^{12} T^{14} + 6505439011133 p^{18} T^{16} - 3643178 p^{24} T^{18} + p^{30} T^{20} \)
97 \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
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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

−8.357650192784133268869138289171, −8.248769012487150467291391348964, −8.013605696082205649768765020321, −7.74657942407784517463432430670, −7.43192695854085614181681259807, −7.25148595476539696283598629308, −6.92156721859609382299166795991, −6.86678798802957610728685683430, −6.74697335752028892035439759222, −6.58253413057200100532441752179, −6.47822961820532445296571330633, −5.91305368120038529267052862920, −5.62036500732862361628891850858, −5.60259368821300895672858813444, −5.47840845349865906440710770840, −5.31112394515550259827808581077, −4.70959041005156714992572015765, −4.40303099004854676081851310537, −4.23731031807583349022840487675, −3.81746618896785242225556130638, −3.60641650265063205159219435507, −3.06015101561518229796816170911, −2.72333088463365132349406283853, −2.25265769513463768454853155610, −1.22196100417709754142613348311, 1.22196100417709754142613348311, 2.25265769513463768454853155610, 2.72333088463365132349406283853, 3.06015101561518229796816170911, 3.60641650265063205159219435507, 3.81746618896785242225556130638, 4.23731031807583349022840487675, 4.40303099004854676081851310537, 4.70959041005156714992572015765, 5.31112394515550259827808581077, 5.47840845349865906440710770840, 5.60259368821300895672858813444, 5.62036500732862361628891850858, 5.91305368120038529267052862920, 6.47822961820532445296571330633, 6.58253413057200100532441752179, 6.74697335752028892035439759222, 6.86678798802957610728685683430, 6.92156721859609382299166795991, 7.25148595476539696283598629308, 7.43192695854085614181681259807, 7.74657942407784517463432430670, 8.013605696082205649768765020321, 8.248769012487150467291391348964, 8.357650192784133268869138289171

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

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