Properties

Label 20-690e10-1.1-c1e10-0-1
Degree $20$
Conductor $2.446\times 10^{28}$
Sign $1$
Analytic cond. $2.57787\times 10^{7}$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 5-s + 6-s − 10-s − 2·11-s − 15-s + 16·17-s + 18·19-s + 2·22-s − 23-s + 22·29-s + 30-s + 8·31-s + 2·33-s − 16·34-s − 16·37-s − 18·38-s − 9·41-s + 2·43-s + 46-s − 48·47-s + 7·49-s − 16·51-s + 2·53-s − 2·55-s − 18·57-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.316·10-s − 0.603·11-s − 0.258·15-s + 3.88·17-s + 4.12·19-s + 0.426·22-s − 0.208·23-s + 4.08·29-s + 0.182·30-s + 1.43·31-s + 0.348·33-s − 2.74·34-s − 2.63·37-s − 2.91·38-s − 1.40·41-s + 0.304·43-s + 0.147·46-s − 7.00·47-s + 49-s − 2.24·51-s + 0.274·53-s − 0.269·55-s − 2.38·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(2.57787\times 10^{7}\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.252462067\)
\(L(\frac12)\) \(\approx\) \(4.252462067\)
\(L(\frac{3}{2})\) 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 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
5 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
23 \( 1 + T - 21 T^{2} - 131 T^{3} + p^{2} T^{4} + 3279 T^{5} + p^{3} T^{6} - 131 p^{2} T^{7} - 21 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
good7 \( 1 - p T^{2} - 11 T^{3} - 6 T^{4} + 143 T^{5} + 4 p^{2} T^{6} + 33 T^{7} - 2087 T^{8} - 4587 T^{9} + 20505 T^{10} - 4587 p T^{11} - 2087 p^{2} T^{12} + 33 p^{3} T^{13} + 4 p^{6} T^{14} + 143 p^{5} T^{15} - 6 p^{6} T^{16} - 11 p^{7} T^{17} - p^{9} T^{18} + p^{10} T^{20} \)
11 \( 1 + 2 T + 4 T^{2} - 14 T^{3} + 38 T^{4} + 318 T^{5} + 515 T^{6} - 4074 T^{7} - 19632 T^{8} - 742 T^{9} + 121309 T^{10} - 742 p T^{11} - 19632 p^{2} T^{12} - 4074 p^{3} T^{13} + 515 p^{4} T^{14} + 318 p^{5} T^{15} + 38 p^{6} T^{16} - 14 p^{7} T^{17} + 4 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 + 20 T^{2} + 33 T^{3} + 59 T^{4} + 121 T^{5} + 1983 T^{6} - 6512 T^{7} + 52959 T^{8} + 95722 T^{9} + 534381 T^{10} + 95722 p T^{11} + 52959 p^{2} T^{12} - 6512 p^{3} T^{13} + 1983 p^{4} T^{14} + 121 p^{5} T^{15} + 59 p^{6} T^{16} + 33 p^{7} T^{17} + 20 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 16 T + 140 T^{2} - 1066 T^{3} + 7427 T^{4} - 44775 T^{5} + 245203 T^{6} - 1270689 T^{7} + 361357 p T^{8} - 27627072 T^{9} + 117076917 T^{10} - 27627072 p T^{11} + 361357 p^{3} T^{12} - 1270689 p^{3} T^{13} + 245203 p^{4} T^{14} - 44775 p^{5} T^{15} + 7427 p^{6} T^{16} - 1066 p^{7} T^{17} + 140 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 18 T + 118 T^{2} - 88 T^{3} - 3474 T^{4} + 23152 T^{5} - 52399 T^{6} - 133826 T^{7} + 1238956 T^{8} - 2877628 T^{9} + 3057009 T^{10} - 2877628 p T^{11} + 1238956 p^{2} T^{12} - 133826 p^{3} T^{13} - 52399 p^{4} T^{14} + 23152 p^{5} T^{15} - 3474 p^{6} T^{16} - 88 p^{7} T^{17} + 118 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 22 T + 191 T^{2} - 902 T^{3} + 6121 T^{4} - 74118 T^{5} + 567301 T^{6} - 2465452 T^{7} + 10926061 T^{8} - 95996714 T^{9} + 682172371 T^{10} - 95996714 p T^{11} + 10926061 p^{2} T^{12} - 2465452 p^{3} T^{13} + 567301 p^{4} T^{14} - 74118 p^{5} T^{15} + 6121 p^{6} T^{16} - 902 p^{7} T^{17} + 191 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 8 T - 77 T^{2} + 886 T^{3} + 29 T^{4} - 24882 T^{5} + 83449 T^{6} - 415120 T^{7} + 3132371 T^{8} + 17771486 T^{9} - 271335241 T^{10} + 17771486 p T^{11} + 3132371 p^{2} T^{12} - 415120 p^{3} T^{13} + 83449 p^{4} T^{14} - 24882 p^{5} T^{15} + 29 p^{6} T^{16} + 886 p^{7} T^{17} - 77 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 16 T + 65 T^{2} - 487 T^{3} - 5929 T^{4} - 16136 T^{5} + 81295 T^{6} + 566191 T^{7} + 105674 T^{8} + 3241018 T^{9} + 86604717 T^{10} + 3241018 p T^{11} + 105674 p^{2} T^{12} + 566191 p^{3} T^{13} + 81295 p^{4} T^{14} - 16136 p^{5} T^{15} - 5929 p^{6} T^{16} - 487 p^{7} T^{17} + 65 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 9 T - 37 T^{2} - 449 T^{3} + 974 T^{4} + 2810 T^{5} - 82514 T^{6} + 360799 T^{7} + 5532014 T^{8} - 7329839 T^{9} - 241204127 T^{10} - 7329839 p T^{11} + 5532014 p^{2} T^{12} + 360799 p^{3} T^{13} - 82514 p^{4} T^{14} + 2810 p^{5} T^{15} + 974 p^{6} T^{16} - 449 p^{7} T^{17} - 37 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 2 T - 6 T^{2} + 274 T^{3} + 733 T^{4} - 17318 T^{5} + 21454 T^{6} + 684705 T^{7} - 6669888 T^{8} - 15012549 T^{9} + 160060669 T^{10} - 15012549 p T^{11} - 6669888 p^{2} T^{12} + 684705 p^{3} T^{13} + 21454 p^{4} T^{14} - 17318 p^{5} T^{15} + 733 p^{6} T^{16} + 274 p^{7} T^{17} - 6 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
47 \( ( 1 + 24 T + 417 T^{2} + 4829 T^{3} + 46238 T^{4} + 343251 T^{5} + 46238 p T^{6} + 4829 p^{2} T^{7} + 417 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
53 \( 1 - 2 T + 72 T^{2} + 864 T^{3} + 6468 T^{4} + 48874 T^{5} + 689115 T^{6} + 6683342 T^{7} + 30043206 T^{8} + 348928690 T^{9} + 3038545709 T^{10} + 348928690 p T^{11} + 30043206 p^{2} T^{12} + 6683342 p^{3} T^{13} + 689115 p^{4} T^{14} + 48874 p^{5} T^{15} + 6468 p^{6} T^{16} + 864 p^{7} T^{17} + 72 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 + 22 T + 403 T^{2} + 5214 T^{3} + 65950 T^{4} + 694408 T^{5} + 7225704 T^{6} + 66511148 T^{7} + 610799522 T^{8} + 5004050986 T^{9} + 40545171897 T^{10} + 5004050986 p T^{11} + 610799522 p^{2} T^{12} + 66511148 p^{3} T^{13} + 7225704 p^{4} T^{14} + 694408 p^{5} T^{15} + 65950 p^{6} T^{16} + 5214 p^{7} T^{17} + 403 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 13 T + 174 T^{2} - 1150 T^{3} + 5172 T^{4} + 17797 T^{5} - 163690 T^{6} + 1237911 T^{7} + 31220285 T^{8} - 438706187 T^{9} + 4524039981 T^{10} - 438706187 p T^{11} + 31220285 p^{2} T^{12} + 1237911 p^{3} T^{13} - 163690 p^{4} T^{14} + 17797 p^{5} T^{15} + 5172 p^{6} T^{16} - 1150 p^{7} T^{17} + 174 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 2 T + 69 T^{2} - 1192 T^{3} + 10202 T^{4} - 105672 T^{5} + 1288756 T^{6} - 9417350 T^{7} + 114561852 T^{8} - 888086704 T^{9} + 6683292341 T^{10} - 888086704 p T^{11} + 114561852 p^{2} T^{12} - 9417350 p^{3} T^{13} + 1288756 p^{4} T^{14} - 105672 p^{5} T^{15} + 10202 p^{6} T^{16} - 1192 p^{7} T^{17} + 69 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 45 T + 1019 T^{2} + 15809 T^{3} + 191147 T^{4} + 1868725 T^{5} + 14630386 T^{6} + 87425609 T^{7} + 324012350 T^{8} - 349975978 T^{9} - 13511487857 T^{10} - 349975978 p T^{11} + 324012350 p^{2} T^{12} + 87425609 p^{3} T^{13} + 14630386 p^{4} T^{14} + 1868725 p^{5} T^{15} + 191147 p^{6} T^{16} + 15809 p^{7} T^{17} + 1019 p^{8} T^{18} + 45 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 21 T + 192 T^{2} - 200 T^{3} - 26 T^{4} - 95866 T^{5} + 1486820 T^{6} - 4000160 T^{7} + 3205729 T^{8} - 5232577 p T^{9} + 7996468776 T^{10} - 5232577 p^{2} T^{11} + 3205729 p^{2} T^{12} - 4000160 p^{3} T^{13} + 1486820 p^{4} T^{14} - 95866 p^{5} T^{15} - 26 p^{6} T^{16} - 200 p^{7} T^{17} + 192 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 66 T + 2000 T^{2} - 36773 T^{3} + 454007 T^{4} - 3911908 T^{5} + 24386860 T^{6} - 145433211 T^{7} + 1599046014 T^{8} - 22626338526 T^{9} + 241754488339 T^{10} - 22626338526 p T^{11} + 1599046014 p^{2} T^{12} - 145433211 p^{3} T^{13} + 24386860 p^{4} T^{14} - 3911908 p^{5} T^{15} + 454007 p^{6} T^{16} - 36773 p^{7} T^{17} + 2000 p^{8} T^{18} - 66 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 15 T + 76 T^{2} + 215 T^{3} - 16903 T^{4} + 286278 T^{5} - 2363947 T^{6} + 11789277 T^{7} + 74894180 T^{8} - 2283604213 T^{9} + 25800177657 T^{10} - 2283604213 p T^{11} + 74894180 p^{2} T^{12} + 11789277 p^{3} T^{13} - 2363947 p^{4} T^{14} + 286278 p^{5} T^{15} - 16903 p^{6} T^{16} + 215 p^{7} T^{17} + 76 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 29 T + 422 T^{2} - 4674 T^{3} + 53273 T^{4} - 633139 T^{5} + 7516692 T^{6} - 86252258 T^{7} + 881132108 T^{8} - 7876456500 T^{9} + 70139498001 T^{10} - 7876456500 p T^{11} + 881132108 p^{2} T^{12} - 86252258 p^{3} T^{13} + 7516692 p^{4} T^{14} - 633139 p^{5} T^{15} + 53273 p^{6} T^{16} - 4674 p^{7} T^{17} + 422 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 27 T + 82 T^{2} - 4893 T^{3} - 54947 T^{4} + 204572 T^{5} + 6328057 T^{6} + 11481365 T^{7} - 355784868 T^{8} - 730216879 T^{9} + 23364821711 T^{10} - 730216879 p T^{11} - 355784868 p^{2} T^{12} + 11481365 p^{3} T^{13} + 6328057 p^{4} T^{14} + 204572 p^{5} T^{15} - 54947 p^{6} T^{16} - 4893 p^{7} T^{17} + 82 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \)
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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

−3.82647374496321100447540230125, −3.64255835008033932619306108341, −3.62456409946719180977361681466, −3.30177395821210730853761444833, −3.24667329754728095767005371131, −3.21241850870509334642916699065, −3.18685354945815921109344342830, −3.12427718685936586183511657408, −3.09791196262011265148873827446, −2.93183909595906995640760446956, −2.82628580361362527783526738646, −2.61196855749864473853930794896, −2.36149303003620916572428548927, −2.30609512678866969139261479966, −2.08600947254292948473093363880, −1.90620578185248865346761045447, −1.66383038167471166228286266937, −1.43773853555935490573050878455, −1.41357295767931540824815225854, −1.38367985764901053381195913837, −1.27993566496376174169907844756, −0.858439599755470623206362895117, −0.830002667289150858411505779126, −0.58258110493233716913466671016, −0.30153751580928731043466560585, 0.30153751580928731043466560585, 0.58258110493233716913466671016, 0.830002667289150858411505779126, 0.858439599755470623206362895117, 1.27993566496376174169907844756, 1.38367985764901053381195913837, 1.41357295767931540824815225854, 1.43773853555935490573050878455, 1.66383038167471166228286266937, 1.90620578185248865346761045447, 2.08600947254292948473093363880, 2.30609512678866969139261479966, 2.36149303003620916572428548927, 2.61196855749864473853930794896, 2.82628580361362527783526738646, 2.93183909595906995640760446956, 3.09791196262011265148873827446, 3.12427718685936586183511657408, 3.18685354945815921109344342830, 3.21241850870509334642916699065, 3.24667329754728095767005371131, 3.30177395821210730853761444833, 3.62456409946719180977361681466, 3.64255835008033932619306108341, 3.82647374496321100447540230125

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

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