Properties

Label 20-6027e10-1.1-c1e10-0-1
Degree $20$
Conductor $6.324\times 10^{37}$
Sign $1$
Analytic cond. $6.66473\times 10^{16}$
Root an. cond. $6.93727$
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  + 4·2-s + 10·3-s + 7·4-s + 6·5-s + 40·6-s + 8·8-s + 55·9-s + 24·10-s − 2·11-s + 70·12-s + 60·15-s + 3·16-s + 8·17-s + 220·18-s + 6·19-s + 42·20-s − 8·22-s + 80·24-s − 2·25-s + 220·27-s + 16·29-s + 240·30-s + 2·31-s − 10·32-s − 20·33-s + 32·34-s + 385·36-s + ⋯
L(s)  = 1  + 2.82·2-s + 5.77·3-s + 7/2·4-s + 2.68·5-s + 16.3·6-s + 2.82·8-s + 55/3·9-s + 7.58·10-s − 0.603·11-s + 20.2·12-s + 15.4·15-s + 3/4·16-s + 1.94·17-s + 51.8·18-s + 1.37·19-s + 9.39·20-s − 1.70·22-s + 16.3·24-s − 2/5·25-s + 42.3·27-s + 2.97·29-s + 43.8·30-s + 0.359·31-s − 1.76·32-s − 3.48·33-s + 5.48·34-s + 64.1·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{10} \cdot 7^{20} \cdot 41^{10}\)
Sign: $1$
Analytic conductor: \(6.66473\times 10^{16}\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{10} \cdot 7^{20} \cdot 41^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24517.69610\)
\(L(\frac12)\) \(\approx\) \(24517.69610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{10} \)
7 \( 1 \)
41 \( ( 1 - T )^{10} \)
good2 \( 1 - p^{2} T + 9 T^{2} - p^{4} T^{3} + 15 p T^{4} - 29 p T^{5} + 23 p^{2} T^{6} - 59 p T^{7} + 153 T^{8} - 15 p^{4} T^{9} + 371 T^{10} - 15 p^{5} T^{11} + 153 p^{2} T^{12} - 59 p^{4} T^{13} + 23 p^{6} T^{14} - 29 p^{6} T^{15} + 15 p^{7} T^{16} - p^{11} T^{17} + 9 p^{8} T^{18} - p^{11} T^{19} + p^{10} T^{20} \)
5 \( 1 - 6 T + 38 T^{2} - 164 T^{3} + 659 T^{4} - 2224 T^{5} + 7004 T^{6} - 19706 T^{7} + 52533 T^{8} - 127784 T^{9} + 59682 p T^{10} - 127784 p T^{11} + 52533 p^{2} T^{12} - 19706 p^{3} T^{13} + 7004 p^{4} T^{14} - 2224 p^{5} T^{15} + 659 p^{6} T^{16} - 164 p^{7} T^{17} + 38 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + 2 T + 6 p T^{2} + 134 T^{3} + 2157 T^{4} + 4472 T^{5} + 4264 p T^{6} + 95528 T^{7} + 68622 p T^{8} + 1433532 T^{9} + 9387500 T^{10} + 1433532 p T^{11} + 68622 p^{3} T^{12} + 95528 p^{3} T^{13} + 4264 p^{5} T^{14} + 4472 p^{5} T^{15} + 2157 p^{6} T^{16} + 134 p^{7} T^{17} + 6 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 + 48 T^{2} + 28 T^{3} + 1199 T^{4} + 1532 T^{5} + 1834 p T^{6} + 3090 p T^{7} + 393645 T^{8} + 54066 p T^{9} + 5445282 T^{10} + 54066 p^{2} T^{11} + 393645 p^{2} T^{12} + 3090 p^{4} T^{13} + 1834 p^{5} T^{14} + 1532 p^{5} T^{15} + 1199 p^{6} T^{16} + 28 p^{7} T^{17} + 48 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 8 T + 120 T^{2} - 852 T^{3} + 7057 T^{4} - 43100 T^{5} + 264424 T^{6} - 1385816 T^{7} + 6987690 T^{8} - 31682976 T^{9} + 136878248 T^{10} - 31682976 p T^{11} + 6987690 p^{2} T^{12} - 1385816 p^{3} T^{13} + 264424 p^{4} T^{14} - 43100 p^{5} T^{15} + 7057 p^{6} T^{16} - 852 p^{7} T^{17} + 120 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 6 T + 102 T^{2} - 16 p T^{3} + 3613 T^{4} - 1794 T^{5} + 76528 T^{6} + 87854 T^{7} + 1987550 T^{8} + 698270 T^{9} + 48206668 T^{10} + 698270 p T^{11} + 1987550 p^{2} T^{12} + 87854 p^{3} T^{13} + 76528 p^{4} T^{14} - 1794 p^{5} T^{15} + 3613 p^{6} T^{16} - 16 p^{8} T^{17} + 102 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 98 T^{2} + 144 T^{3} + 5475 T^{4} + 13058 T^{5} + 220138 T^{6} + 27154 p T^{7} + 6998047 T^{8} + 19993516 T^{9} + 178373314 T^{10} + 19993516 p T^{11} + 6998047 p^{2} T^{12} + 27154 p^{4} T^{13} + 220138 p^{4} T^{14} + 13058 p^{5} T^{15} + 5475 p^{6} T^{16} + 144 p^{7} T^{17} + 98 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 - 16 T + 310 T^{2} - 3516 T^{3} + 40487 T^{4} - 357574 T^{5} + 3077532 T^{6} - 763726 p T^{7} + 153739717 T^{8} - 921608216 T^{9} + 5317856190 T^{10} - 921608216 p T^{11} + 153739717 p^{2} T^{12} - 763726 p^{4} T^{13} + 3077532 p^{4} T^{14} - 357574 p^{5} T^{15} + 40487 p^{6} T^{16} - 3516 p^{7} T^{17} + 310 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 2 T + 116 T^{2} + 92 T^{3} + 191 p T^{4} + 19890 T^{5} + 248332 T^{6} + 967122 T^{7} + 11053138 T^{8} + 27524046 T^{9} + 407547432 T^{10} + 27524046 p T^{11} + 11053138 p^{2} T^{12} + 967122 p^{3} T^{13} + 248332 p^{4} T^{14} + 19890 p^{5} T^{15} + 191 p^{7} T^{16} + 92 p^{7} T^{17} + 116 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 24 T + 538 T^{2} - 7792 T^{3} + 104815 T^{4} - 1117896 T^{5} + 11149916 T^{6} - 94165232 T^{7} + 747504257 T^{8} - 5157365748 T^{9} + 33528755634 T^{10} - 5157365748 p T^{11} + 747504257 p^{2} T^{12} - 94165232 p^{3} T^{13} + 11149916 p^{4} T^{14} - 1117896 p^{5} T^{15} + 104815 p^{6} T^{16} - 7792 p^{7} T^{17} + 538 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 8 T + 218 T^{2} - 32 p T^{3} + 23425 T^{4} - 132920 T^{5} + 1737564 T^{6} - 9134876 T^{7} + 99001618 T^{8} - 484688764 T^{9} + 4634185980 T^{10} - 484688764 p T^{11} + 99001618 p^{2} T^{12} - 9134876 p^{3} T^{13} + 1737564 p^{4} T^{14} - 132920 p^{5} T^{15} + 23425 p^{6} T^{16} - 32 p^{8} T^{17} + 218 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 8 T + 366 T^{2} + 2570 T^{3} + 62333 T^{4} + 390010 T^{5} + 6614724 T^{6} + 36981744 T^{7} + 490133401 T^{8} + 2423132064 T^{9} + 26678652750 T^{10} + 2423132064 p T^{11} + 490133401 p^{2} T^{12} + 36981744 p^{3} T^{13} + 6614724 p^{4} T^{14} + 390010 p^{5} T^{15} + 62333 p^{6} T^{16} + 2570 p^{7} T^{17} + 366 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 24 T + 514 T^{2} - 7898 T^{3} + 107415 T^{4} - 1238542 T^{5} + 13026696 T^{6} - 122579350 T^{7} + 1073886733 T^{8} - 8629013754 T^{9} + 65348107198 T^{10} - 8629013754 p T^{11} + 1073886733 p^{2} T^{12} - 122579350 p^{3} T^{13} + 13026696 p^{4} T^{14} - 1238542 p^{5} T^{15} + 107415 p^{6} T^{16} - 7898 p^{7} T^{17} + 514 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 6 T + 378 T^{2} - 1746 T^{3} + 67989 T^{4} - 242568 T^{5} + 7873660 T^{6} - 21845364 T^{7} + 665942870 T^{8} - 1504966448 T^{9} + 43966658252 T^{10} - 1504966448 p T^{11} + 665942870 p^{2} T^{12} - 21845364 p^{3} T^{13} + 7873660 p^{4} T^{14} - 242568 p^{5} T^{15} + 67989 p^{6} T^{16} - 1746 p^{7} T^{17} + 378 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 14 T + 536 T^{2} + 6308 T^{3} + 131969 T^{4} + 1322790 T^{5} + 19754184 T^{6} + 169831458 T^{7} + 2001213738 T^{8} + 14747478254 T^{9} + 143891911336 T^{10} + 14747478254 p T^{11} + 2001213738 p^{2} T^{12} + 169831458 p^{3} T^{13} + 19754184 p^{4} T^{14} + 1322790 p^{5} T^{15} + 131969 p^{6} T^{16} + 6308 p^{7} T^{17} + 536 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 26 T + 514 T^{2} - 122 p T^{3} + 116745 T^{4} - 1455930 T^{5} + 16793248 T^{6} - 176220266 T^{7} + 1728203993 T^{8} - 15635115540 T^{9} + 132956617438 T^{10} - 15635115540 p T^{11} + 1728203993 p^{2} T^{12} - 176220266 p^{3} T^{13} + 16793248 p^{4} T^{14} - 1455930 p^{5} T^{15} + 116745 p^{6} T^{16} - 122 p^{8} T^{17} + 514 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 14 T + 534 T^{2} - 5672 T^{3} + 129393 T^{4} - 1150158 T^{5} + 20009156 T^{6} - 154067206 T^{7} + 2202026914 T^{8} - 14767316402 T^{9} + 180007114196 T^{10} - 14767316402 p T^{11} + 2202026914 p^{2} T^{12} - 154067206 p^{3} T^{13} + 20009156 p^{4} T^{14} - 1150158 p^{5} T^{15} + 129393 p^{6} T^{16} - 5672 p^{7} T^{17} + 534 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 + 36 T + 880 T^{2} + 16122 T^{3} + 254133 T^{4} + 3488554 T^{5} + 43501492 T^{6} + 490808606 T^{7} + 5111306958 T^{8} + 48867019578 T^{9} + 434573845280 T^{10} + 48867019578 p T^{11} + 5111306958 p^{2} T^{12} + 490808606 p^{3} T^{13} + 43501492 p^{4} T^{14} + 3488554 p^{5} T^{15} + 254133 p^{6} T^{16} + 16122 p^{7} T^{17} + 880 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 20 T + 658 T^{2} - 10838 T^{3} + 209749 T^{4} - 2835350 T^{5} + 40998092 T^{6} - 464639668 T^{7} + 5421072693 T^{8} - 51989761152 T^{9} + 506521052030 T^{10} - 51989761152 p T^{11} + 5421072693 p^{2} T^{12} - 464639668 p^{3} T^{13} + 40998092 p^{4} T^{14} - 2835350 p^{5} T^{15} + 209749 p^{6} T^{16} - 10838 p^{7} T^{17} + 658 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 40 T + 1374 T^{2} - 31432 T^{3} + 638549 T^{4} - 10452320 T^{5} + 155683176 T^{6} - 1989519488 T^{7} + 23479587026 T^{8} - 244087047392 T^{9} + 2360303417268 T^{10} - 244087047392 p T^{11} + 23479587026 p^{2} T^{12} - 1989519488 p^{3} T^{13} + 155683176 p^{4} T^{14} - 10452320 p^{5} T^{15} + 638549 p^{6} T^{16} - 31432 p^{7} T^{17} + 1374 p^{8} T^{18} - 40 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 2 T + 378 T^{2} - 286 T^{3} + 63741 T^{4} - 10196 T^{5} + 6432776 T^{6} - 4935452 T^{7} + 479133858 T^{8} - 1262682528 T^{9} + 36809593756 T^{10} - 1262682528 p T^{11} + 479133858 p^{2} T^{12} - 4935452 p^{3} T^{13} + 6432776 p^{4} T^{14} - 10196 p^{5} T^{15} + 63741 p^{6} T^{16} - 286 p^{7} T^{17} + 378 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 16 T + 692 T^{2} - 10070 T^{3} + 232851 T^{4} - 3037760 T^{5} + 50145138 T^{6} - 581263526 T^{7} + 7634856185 T^{8} - 77833193648 T^{9} + 857204295090 T^{10} - 77833193648 p T^{11} + 7634856185 p^{2} T^{12} - 581263526 p^{3} T^{13} + 50145138 p^{4} T^{14} - 3037760 p^{5} T^{15} + 232851 p^{6} T^{16} - 10070 p^{7} T^{17} + 692 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \)
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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

−2.69091344417579167122720727013, −2.62435508923111055157110840280, −2.60108701532947103447280950779, −2.49899218884091392790966845723, −2.42396186848188879689290221037, −2.32565503584429645333259466034, −2.32083609452337299397833506030, −2.28165866718398723951305143508, −2.12176741156728473396460737470, −2.11365215620953995669212842461, −1.96297413144694461362702429108, −1.90119720551076932124974462888, −1.76051224045935011022402016586, −1.71552293201010633308245277380, −1.66180237912596824931695450130, −1.43001840292983827356598177518, −1.30534341464179241671706084790, −1.20879598216017996466507503984, −0.972047485280790649341909945447, −0.906462126141152927843464712011, −0.892052354939303416293387382083, −0.806581944972533613017313207199, −0.68816547974548875037957114882, −0.52184376373507037061603481761, −0.38571943161191844602192312708, 0.38571943161191844602192312708, 0.52184376373507037061603481761, 0.68816547974548875037957114882, 0.806581944972533613017313207199, 0.892052354939303416293387382083, 0.906462126141152927843464712011, 0.972047485280790649341909945447, 1.20879598216017996466507503984, 1.30534341464179241671706084790, 1.43001840292983827356598177518, 1.66180237912596824931695450130, 1.71552293201010633308245277380, 1.76051224045935011022402016586, 1.90119720551076932124974462888, 1.96297413144694461362702429108, 2.11365215620953995669212842461, 2.12176741156728473396460737470, 2.28165866718398723951305143508, 2.32083609452337299397833506030, 2.32565503584429645333259466034, 2.42396186848188879689290221037, 2.49899218884091392790966845723, 2.60108701532947103447280950779, 2.62435508923111055157110840280, 2.69091344417579167122720727013

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

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