Properties

Label 20-4004e10-1.1-c1e10-0-1
Degree $20$
Conductor $1.059\times 10^{36}$
Sign $1$
Analytic cond. $1.11612\times 10^{15}$
Root an. cond. $5.65438$
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  + 3-s + 4·5-s − 10·7-s − 6·9-s + 10·11-s + 10·13-s + 4·15-s + 3·17-s + 6·19-s − 10·21-s + 4·23-s − 6·25-s − 9·27-s + 10·29-s − 31-s + 10·33-s − 40·35-s + 20·37-s + 10·39-s − 24·45-s + 4·47-s + 55·49-s + 3·51-s − 5·53-s + 40·55-s + 6·57-s + 11·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s − 3.77·7-s − 2·9-s + 3.01·11-s + 2.77·13-s + 1.03·15-s + 0.727·17-s + 1.37·19-s − 2.18·21-s + 0.834·23-s − 6/5·25-s − 1.73·27-s + 1.85·29-s − 0.179·31-s + 1.74·33-s − 6.76·35-s + 3.28·37-s + 1.60·39-s − 3.57·45-s + 0.583·47-s + 55/7·49-s + 0.420·51-s − 0.686·53-s + 5.39·55-s + 0.794·57-s + 1.43·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(1.11612\times 10^{15}\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(96.64882360\)
\(L(\frac12)\) \(\approx\) \(96.64882360\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + T )^{10} \)
11 \( ( 1 - T )^{10} \)
13 \( ( 1 - T )^{10} \)
good3 \( 1 - T + 7 T^{2} - 4 T^{3} + 23 T^{4} - 2 p T^{5} + 31 p T^{6} - 4 p^{2} T^{7} + 121 p T^{8} - 43 p T^{9} + 362 p T^{10} - 43 p^{2} T^{11} + 121 p^{3} T^{12} - 4 p^{5} T^{13} + 31 p^{5} T^{14} - 2 p^{6} T^{15} + 23 p^{6} T^{16} - 4 p^{7} T^{17} + 7 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
5 \( 1 - 4 T + 22 T^{2} - 66 T^{3} + 244 T^{4} - 639 T^{5} + 2066 T^{6} - 4878 T^{7} + 13658 T^{8} - 28914 T^{9} + 73054 T^{10} - 28914 p T^{11} + 13658 p^{2} T^{12} - 4878 p^{3} T^{13} + 2066 p^{4} T^{14} - 639 p^{5} T^{15} + 244 p^{6} T^{16} - 66 p^{7} T^{17} + 22 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 3 T + 67 T^{2} - 106 T^{3} + 1907 T^{4} - 606 T^{5} + 39625 T^{6} - 208 T^{7} + 923147 T^{8} - 619153 T^{9} + 1110290 p T^{10} - 619153 p T^{11} + 923147 p^{2} T^{12} - 208 p^{3} T^{13} + 39625 p^{4} T^{14} - 606 p^{5} T^{15} + 1907 p^{6} T^{16} - 106 p^{7} T^{17} + 67 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 6 T + 106 T^{2} - 536 T^{3} + 5848 T^{4} - 26069 T^{5} + 217866 T^{6} - 857970 T^{7} + 6003504 T^{8} - 21091948 T^{9} + 128702906 T^{10} - 21091948 p T^{11} + 6003504 p^{2} T^{12} - 857970 p^{3} T^{13} + 217866 p^{4} T^{14} - 26069 p^{5} T^{15} + 5848 p^{6} T^{16} - 536 p^{7} T^{17} + 106 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 - 4 T + 89 T^{2} - 568 T^{3} + 4642 T^{4} - 34097 T^{5} + 200694 T^{6} - 1267095 T^{7} + 7066605 T^{8} - 35247826 T^{9} + 188451394 T^{10} - 35247826 p T^{11} + 7066605 p^{2} T^{12} - 1267095 p^{3} T^{13} + 200694 p^{4} T^{14} - 34097 p^{5} T^{15} + 4642 p^{6} T^{16} - 568 p^{7} T^{17} + 89 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 10 T + 189 T^{2} - 1474 T^{3} + 16002 T^{4} - 102647 T^{5} + 843184 T^{6} - 4679811 T^{7} + 32667181 T^{8} - 163727734 T^{9} + 1030736646 T^{10} - 163727734 p T^{11} + 32667181 p^{2} T^{12} - 4679811 p^{3} T^{13} + 843184 p^{4} T^{14} - 102647 p^{5} T^{15} + 16002 p^{6} T^{16} - 1474 p^{7} T^{17} + 189 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + T + 132 T^{2} + 97 T^{3} + 9474 T^{4} + 7035 T^{5} + 492894 T^{6} + 437061 T^{7} + 20131597 T^{8} + 18540970 T^{9} + 678680924 T^{10} + 18540970 p T^{11} + 20131597 p^{2} T^{12} + 437061 p^{3} T^{13} + 492894 p^{4} T^{14} + 7035 p^{5} T^{15} + 9474 p^{6} T^{16} + 97 p^{7} T^{17} + 132 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 20 T + 356 T^{2} - 4035 T^{3} + 39174 T^{4} - 276725 T^{5} + 1549834 T^{6} - 4392123 T^{7} - 11642247 T^{8} + 313449567 T^{9} - 2332182844 T^{10} + 313449567 p T^{11} - 11642247 p^{2} T^{12} - 4392123 p^{3} T^{13} + 1549834 p^{4} T^{14} - 276725 p^{5} T^{15} + 39174 p^{6} T^{16} - 4035 p^{7} T^{17} + 356 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 64 T^{2} - 281 T^{3} + 4670 T^{4} - 11107 T^{5} + 306890 T^{6} - 633177 T^{7} + 15457121 T^{8} - 40876763 T^{9} + 624010988 T^{10} - 40876763 p T^{11} + 15457121 p^{2} T^{12} - 633177 p^{3} T^{13} + 306890 p^{4} T^{14} - 11107 p^{5} T^{15} + 4670 p^{6} T^{16} - 281 p^{7} T^{17} + 64 p^{8} T^{18} + p^{10} T^{20} \)
43 \( 1 + 193 T^{2} - 408 T^{3} + 22210 T^{4} - 56277 T^{5} + 1831614 T^{6} - 5104952 T^{7} + 112508301 T^{8} - 302767828 T^{9} + 5498399520 T^{10} - 302767828 p T^{11} + 112508301 p^{2} T^{12} - 5104952 p^{3} T^{13} + 1831614 p^{4} T^{14} - 56277 p^{5} T^{15} + 22210 p^{6} T^{16} - 408 p^{7} T^{17} + 193 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 4 T + 7 p T^{2} - 1432 T^{3} + 53170 T^{4} - 232337 T^{5} + 5543238 T^{6} - 22864431 T^{7} + 409961757 T^{8} - 1522478962 T^{9} + 22334479810 T^{10} - 1522478962 p T^{11} + 409961757 p^{2} T^{12} - 22864431 p^{3} T^{13} + 5543238 p^{4} T^{14} - 232337 p^{5} T^{15} + 53170 p^{6} T^{16} - 1432 p^{7} T^{17} + 7 p^{9} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 + 5 T + 376 T^{2} + 1913 T^{3} + 69148 T^{4} + 338693 T^{5} + 8156430 T^{6} + 36950489 T^{7} + 680034810 T^{8} + 2756338605 T^{9} + 41751994840 T^{10} + 2756338605 p T^{11} + 680034810 p^{2} T^{12} + 36950489 p^{3} T^{13} + 8156430 p^{4} T^{14} + 338693 p^{5} T^{15} + 69148 p^{6} T^{16} + 1913 p^{7} T^{17} + 376 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 11 T + 435 T^{2} - 3622 T^{3} + 84214 T^{4} - 562789 T^{5} + 10153794 T^{6} - 56970701 T^{7} + 881350905 T^{8} - 4292587607 T^{9} + 58788179638 T^{10} - 4292587607 p T^{11} + 881350905 p^{2} T^{12} - 56970701 p^{3} T^{13} + 10153794 p^{4} T^{14} - 562789 p^{5} T^{15} + 84214 p^{6} T^{16} - 3622 p^{7} T^{17} + 435 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 12 T + 435 T^{2} - 4649 T^{3} + 90899 T^{4} - 874306 T^{5} + 12155071 T^{6} - 104460383 T^{7} + 1153654937 T^{8} - 8748667246 T^{9} + 81208239906 T^{10} - 8748667246 p T^{11} + 1153654937 p^{2} T^{12} - 104460383 p^{3} T^{13} + 12155071 p^{4} T^{14} - 874306 p^{5} T^{15} + 90899 p^{6} T^{16} - 4649 p^{7} T^{17} + 435 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 2 T + 337 T^{2} + 1558 T^{3} + 54861 T^{4} + 380974 T^{5} + 6238057 T^{6} + 50098140 T^{7} + 576557197 T^{8} + 4423963664 T^{9} + 43368686470 T^{10} + 4423963664 p T^{11} + 576557197 p^{2} T^{12} + 50098140 p^{3} T^{13} + 6238057 p^{4} T^{14} + 380974 p^{5} T^{15} + 54861 p^{6} T^{16} + 1558 p^{7} T^{17} + 337 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 28 T + 561 T^{2} - 7544 T^{3} + 85222 T^{4} - 740510 T^{5} + 5713998 T^{6} - 35344402 T^{7} + 227321625 T^{8} - 1316810464 T^{9} + 11061781762 T^{10} - 1316810464 p T^{11} + 227321625 p^{2} T^{12} - 35344402 p^{3} T^{13} + 5713998 p^{4} T^{14} - 740510 p^{5} T^{15} + 85222 p^{6} T^{16} - 7544 p^{7} T^{17} + 561 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 11 T + 510 T^{2} - 3876 T^{3} + 107427 T^{4} - 503382 T^{5} + 12454094 T^{6} - 21917736 T^{7} + 945186916 T^{8} + 866262141 T^{9} + 64033920600 T^{10} + 866262141 p T^{11} + 945186916 p^{2} T^{12} - 21917736 p^{3} T^{13} + 12454094 p^{4} T^{14} - 503382 p^{5} T^{15} + 107427 p^{6} T^{16} - 3876 p^{7} T^{17} + 510 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 10 T + 419 T^{2} + 5562 T^{3} + 100146 T^{4} + 1304087 T^{5} + 17192560 T^{6} + 193333974 T^{7} + 2112022323 T^{8} + 20791995852 T^{9} + 191448301804 T^{10} + 20791995852 p T^{11} + 2112022323 p^{2} T^{12} + 193333974 p^{3} T^{13} + 17192560 p^{4} T^{14} + 1304087 p^{5} T^{15} + 100146 p^{6} T^{16} + 5562 p^{7} T^{17} + 419 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 7 T + 419 T^{2} - 2713 T^{3} + 92752 T^{4} - 546245 T^{5} + 13858840 T^{6} - 74597083 T^{7} + 1575729275 T^{8} - 7744325731 T^{9} + 144155004454 T^{10} - 7744325731 p T^{11} + 1575729275 p^{2} T^{12} - 74597083 p^{3} T^{13} + 13858840 p^{4} T^{14} - 546245 p^{5} T^{15} + 92752 p^{6} T^{16} - 2713 p^{7} T^{17} + 419 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 30 T + 720 T^{2} - 10616 T^{3} + 133146 T^{4} - 1151417 T^{5} + 8467506 T^{6} - 27649542 T^{7} - 60454374 T^{8} + 4001725886 T^{9} - 38334028578 T^{10} + 4001725886 p T^{11} - 60454374 p^{2} T^{12} - 27649542 p^{3} T^{13} + 8467506 p^{4} T^{14} - 1151417 p^{5} T^{15} + 133146 p^{6} T^{16} - 10616 p^{7} T^{17} + 720 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 55 T + 1700 T^{2} - 38512 T^{3} + 715495 T^{4} - 11557266 T^{5} + 167029752 T^{6} - 2190126020 T^{7} + 26278509800 T^{8} - 290387953875 T^{9} + 2970183426280 T^{10} - 290387953875 p T^{11} + 26278509800 p^{2} T^{12} - 2190126020 p^{3} T^{13} + 167029752 p^{4} T^{14} - 11557266 p^{5} T^{15} + 715495 p^{6} T^{16} - 38512 p^{7} T^{17} + 1700 p^{8} T^{18} - 55 p^{9} T^{19} + p^{10} T^{20} \)
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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

−2.95369078131982193213310803701, −2.90643621970745581581468483330, −2.83237768024002490110537049179, −2.78839274458673937616925098927, −2.69969190790297414351959036474, −2.24512719967444384112371600656, −2.20228159088048397675125234389, −2.19715000658002304645959908327, −2.14281882369186131036977112635, −2.12386423202194109899587921368, −2.01572410152088901564635627189, −1.93950067949716787825757204639, −1.79737790533281673615954118042, −1.66884560252663118255134698947, −1.57526569700133273191591602418, −1.33966880874324325288976718365, −1.12035737187660279653215716639, −0.994226386747620171642476936128, −0.837750622730866174328864606099, −0.819700919020974439616351569158, −0.76334418696860147219945602977, −0.72946324586928882934103896415, −0.68916212851347308564231600474, −0.35626338327080995097521088751, −0.34524810506934720687346253936, 0.34524810506934720687346253936, 0.35626338327080995097521088751, 0.68916212851347308564231600474, 0.72946324586928882934103896415, 0.76334418696860147219945602977, 0.819700919020974439616351569158, 0.837750622730866174328864606099, 0.994226386747620171642476936128, 1.12035737187660279653215716639, 1.33966880874324325288976718365, 1.57526569700133273191591602418, 1.66884560252663118255134698947, 1.79737790533281673615954118042, 1.93950067949716787825757204639, 2.01572410152088901564635627189, 2.12386423202194109899587921368, 2.14281882369186131036977112635, 2.19715000658002304645959908327, 2.20228159088048397675125234389, 2.24512719967444384112371600656, 2.69969190790297414351959036474, 2.78839274458673937616925098927, 2.83237768024002490110537049179, 2.90643621970745581581468483330, 2.95369078131982193213310803701

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

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