Properties

Label 20-39e20-1.1-c3e10-0-0
Degree $20$
Conductor $6.627\times 10^{31}$
Sign $1$
Analytic cond. $3.38807\times 10^{19}$
Root an. cond. $9.47322$
Motivic weight $3$
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  − 10·4-s + 55·16-s − 210·17-s + 120·23-s − 145·25-s − 990·29-s − 740·43-s − 940·49-s − 330·53-s + 2.75e3·61-s − 360·64-s + 2.10e3·68-s + 1.10e3·79-s − 1.20e3·92-s + 1.45e3·100-s + 570·101-s + 6.10e3·103-s − 2.52e3·107-s − 5.07e3·113-s + 9.90e3·116-s − 2.53e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 5/4·4-s + 0.859·16-s − 2.99·17-s + 1.08·23-s − 1.15·25-s − 6.33·29-s − 2.62·43-s − 2.74·49-s − 0.855·53-s + 5.77·61-s − 0.703·64-s + 3.74·68-s + 1.56·79-s − 1.35·92-s + 1.44·100-s + 0.561·101-s + 5.83·103-s − 2.27·107-s − 4.22·113-s + 7.92·116-s − 1.90·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{20} \cdot 13^{20}\)
Sign: $1$
Analytic conductor: \(3.38807\times 10^{19}\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{20} \cdot 13^{20} ,\ ( \ : [3/2]^{10} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.735202385\)
\(L(\frac12)\) \(\approx\) \(2.735202385\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 5 p T^{2} + 45 T^{4} + 65 p^{2} T^{6} + 1505 p^{2} T^{8} + 1017 p^{6} T^{10} + 1505 p^{8} T^{12} + 65 p^{14} T^{14} + 45 p^{18} T^{16} + 5 p^{25} T^{18} + p^{30} T^{20} \)
5 \( 1 + 29 p T^{2} + 8016 p T^{4} + 1274791 p T^{6} + 171848431 p T^{8} + 130789308432 T^{10} + 171848431 p^{7} T^{12} + 1274791 p^{13} T^{14} + 8016 p^{19} T^{16} + 29 p^{25} T^{18} + p^{30} T^{20} \)
7 \( 1 + 940 T^{2} + 771810 T^{4} + 57543450 p T^{6} + 27644677335 p T^{8} + 69042388808964 T^{10} + 27644677335 p^{7} T^{12} + 57543450 p^{13} T^{14} + 771810 p^{18} T^{16} + 940 p^{24} T^{18} + p^{30} T^{20} \)
11 \( 1 + 230 p T^{2} + 2957385 T^{4} + 2242820720 T^{6} + 8608466586110 T^{8} + 16313181047368860 T^{10} + 8608466586110 p^{6} T^{12} + 2242820720 p^{12} T^{14} + 2957385 p^{18} T^{16} + 230 p^{25} T^{18} + p^{30} T^{20} \)
17 \( ( 1 + 105 T + 24010 T^{2} + 1806465 T^{3} + 227765305 T^{4} + 12663211248 T^{5} + 227765305 p^{3} T^{6} + 1806465 p^{6} T^{7} + 24010 p^{9} T^{8} + 105 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
19 \( 1 + 27250 T^{2} + 388244505 T^{4} + 3821333646960 T^{6} + 30269244251252670 T^{8} + \)\(21\!\cdots\!60\)\( T^{10} + 30269244251252670 p^{6} T^{12} + 3821333646960 p^{12} T^{14} + 388244505 p^{18} T^{16} + 27250 p^{24} T^{18} + p^{30} T^{20} \)
23 \( ( 1 - 60 T + 1175 p T^{2} - 2112420 T^{3} + 521121160 T^{4} - 25485650352 T^{5} + 521121160 p^{3} T^{6} - 2112420 p^{6} T^{7} + 1175 p^{10} T^{8} - 60 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
29 \( ( 1 + 495 T + 191860 T^{2} + 49831245 T^{3} + 10880191495 T^{4} + 1841365752984 T^{5} + 10880191495 p^{3} T^{6} + 49831245 p^{6} T^{7} + 191860 p^{9} T^{8} + 495 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
31 \( 1 + 181120 T^{2} + 16624601430 T^{4} + 1013037768252330 T^{6} + 45190749109342797945 T^{8} + \)\(15\!\cdots\!52\)\( T^{10} + 45190749109342797945 p^{6} T^{12} + 1013037768252330 p^{12} T^{14} + 16624601430 p^{18} T^{16} + 181120 p^{24} T^{18} + p^{30} T^{20} \)
37 \( 1 + 269245 T^{2} + 39522875220 T^{4} + 3933194266635915 T^{6} + \)\(29\!\cdots\!95\)\( T^{8} + \)\(16\!\cdots\!44\)\( T^{10} + \)\(29\!\cdots\!95\)\( p^{6} T^{12} + 3933194266635915 p^{12} T^{14} + 39522875220 p^{18} T^{16} + 269245 p^{24} T^{18} + p^{30} T^{20} \)
41 \( 1 + 411805 T^{2} + 86434782780 T^{4} + 12079291529986715 T^{6} + \)\(12\!\cdots\!95\)\( T^{8} + \)\(97\!\cdots\!60\)\( T^{10} + \)\(12\!\cdots\!95\)\( p^{6} T^{12} + 12079291529986715 p^{12} T^{14} + 86434782780 p^{18} T^{16} + 411805 p^{24} T^{18} + p^{30} T^{20} \)
43 \( ( 1 + 370 T + 285930 T^{2} + 2189100 p T^{3} + 42432499695 T^{4} + 10063056645144 T^{5} + 42432499695 p^{3} T^{6} + 2189100 p^{7} T^{7} + 285930 p^{9} T^{8} + 370 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
47 \( 1 + 856570 T^{2} + 341810552865 T^{4} + 84145551757645040 T^{6} + \)\(14\!\cdots\!10\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{10} + \)\(14\!\cdots\!10\)\( p^{6} T^{12} + 84145551757645040 p^{12} T^{14} + 341810552865 p^{18} T^{16} + 856570 p^{24} T^{18} + p^{30} T^{20} \)
53 \( ( 1 + 165 T + 569020 T^{2} + 43686555 T^{3} + 139448077615 T^{4} + 5740341322068 T^{5} + 139448077615 p^{3} T^{6} + 43686555 p^{6} T^{7} + 569020 p^{9} T^{8} + 165 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
59 \( 1 + 1345990 T^{2} + 903767605605 T^{4} + 396684631924387880 T^{6} + \)\(12\!\cdots\!70\)\( T^{8} + \)\(29\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!70\)\( p^{6} T^{12} + 396684631924387880 p^{12} T^{14} + 903767605605 p^{18} T^{16} + 1345990 p^{24} T^{18} + p^{30} T^{20} \)
61 \( ( 1 - 1375 T + 1852065 T^{2} - 1426380500 T^{3} + 1024532648225 T^{4} - 506775366057195 T^{5} + 1024532648225 p^{3} T^{6} - 1426380500 p^{6} T^{7} + 1852065 p^{9} T^{8} - 1375 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
67 \( 1 + 2072020 T^{2} + 2144201110170 T^{4} + 1431538254550407390 T^{6} + \)\(67\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!88\)\( T^{10} + \)\(67\!\cdots\!45\)\( p^{6} T^{12} + 1431538254550407390 p^{12} T^{14} + 2144201110170 p^{18} T^{16} + 2072020 p^{24} T^{18} + p^{30} T^{20} \)
71 \( 1 + 2648890 T^{2} + 3285865351665 T^{4} + 2550853350944883440 T^{6} + \)\(13\!\cdots\!90\)\( T^{8} + \)\(57\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!90\)\( p^{6} T^{12} + 2550853350944883440 p^{12} T^{14} + 3285865351665 p^{18} T^{16} + 2648890 p^{24} T^{18} + p^{30} T^{20} \)
73 \( 1 + 3289555 T^{2} + 4969741417695 T^{4} + 4586689788238803090 T^{6} + \)\(28\!\cdots\!65\)\( T^{8} + \)\(13\!\cdots\!89\)\( T^{10} + \)\(28\!\cdots\!65\)\( p^{6} T^{12} + 4586689788238803090 p^{12} T^{14} + 4969741417695 p^{18} T^{16} + 3289555 p^{24} T^{18} + p^{30} T^{20} \)
79 \( ( 1 - 550 T + 1823820 T^{2} - 1009940540 T^{3} + 1533520086095 T^{4} - 729404155596156 T^{5} + 1533520086095 p^{3} T^{6} - 1009940540 p^{6} T^{7} + 1823820 p^{9} T^{8} - 550 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
83 \( 1 + 2310970 T^{2} + 2288506077705 T^{4} + 1078317863510325200 T^{6} + \)\(72\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{10} + \)\(72\!\cdots\!70\)\( p^{6} T^{12} + 1078317863510325200 p^{12} T^{14} + 2288506077705 p^{18} T^{16} + 2310970 p^{24} T^{18} + p^{30} T^{20} \)
89 \( 1 + 5529130 T^{2} + 14414210525565 T^{4} + 23453015761759771640 T^{6} + \)\(26\!\cdots\!50\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!50\)\( p^{6} T^{12} + 23453015761759771640 p^{12} T^{14} + 14414210525565 p^{18} T^{16} + 5529130 p^{24} T^{18} + p^{30} T^{20} \)
97 \( 1 + 4570300 T^{2} + 11362688693910 T^{4} + 19272697712381739630 T^{6} + \)\(24\!\cdots\!85\)\( T^{8} + \)\(25\!\cdots\!28\)\( T^{10} + \)\(24\!\cdots\!85\)\( p^{6} T^{12} + 19272697712381739630 p^{12} T^{14} + 11362688693910 p^{18} T^{16} + 4570300 p^{24} T^{18} + p^{30} T^{20} \)
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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

−3.03076895333184659764406974593, −2.98224324607426310874782425644, −2.91862361283585892027380121044, −2.52458619512019929331950559126, −2.41592038544018995897247183873, −2.29495925968687375903221503636, −2.25560464040033717588463149438, −2.16752651091106886120045402789, −2.16452718410463033506869377886, −2.00843185242441460475767685929, −1.90265759062467063737005529845, −1.79271479306014364518367995735, −1.67125299763538069694510240836, −1.50020161056778332147490104735, −1.42430141688623934322855723612, −1.38724509636436461059490329200, −1.31529711699910105570533591628, −0.78768445496912214744739203245, −0.77102746119186337735703738319, −0.65912035406129351524050767619, −0.64287455096588829505123079905, −0.35539150661988634740455470305, −0.25675958354240277807929701429, −0.22467691991089648213578642388, −0.18480224497698183056356409812, 0.18480224497698183056356409812, 0.22467691991089648213578642388, 0.25675958354240277807929701429, 0.35539150661988634740455470305, 0.64287455096588829505123079905, 0.65912035406129351524050767619, 0.77102746119186337735703738319, 0.78768445496912214744739203245, 1.31529711699910105570533591628, 1.38724509636436461059490329200, 1.42430141688623934322855723612, 1.50020161056778332147490104735, 1.67125299763538069694510240836, 1.79271479306014364518367995735, 1.90265759062467063737005529845, 2.00843185242441460475767685929, 2.16452718410463033506869377886, 2.16752651091106886120045402789, 2.25560464040033717588463149438, 2.29495925968687375903221503636, 2.41592038544018995897247183873, 2.52458619512019929331950559126, 2.91862361283585892027380121044, 2.98224324607426310874782425644, 3.03076895333184659764406974593

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

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