↑

Properties

Label 20-7644e10-1.1-c1e10-0-1
Degree $20$
Conductor $6.811\times 10^{38}$
Sign $1$
Analytic cond. $7.17757\times 10^{17}$
Root an. cond. $7.81265$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10·3-s + 55·9-s + 13-s − 4·17-s − 8·23-s + 13·25-s + 220·27-s − 2·29-s + 10·39-s − 6·43-s − 40·51-s + 24·53-s − 80·69-s + 130·75-s + 6·79-s + 715·81-s − 20·87-s − 14·101-s + 12·103-s − 24·107-s + 22·113-s + 55·117-s + 28·121-s + 127-s − 60·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 5.77·3-s + 55/3·9-s + 0.277·13-s − 0.970·17-s − 1.66·23-s + 13/5·25-s + 42.3·27-s − 0.371·29-s + 1.60·39-s − 0.914·43-s − 5.60·51-s + 3.29·53-s − 9.63·69-s + 15.0·75-s + 0.675·79-s + 79.4·81-s − 2.14·87-s − 1.39·101-s + 1.18·103-s − 2.32·107-s + 2.06·113-s + 5.08·117-s + 2.54·121-s + 0.0887·127-s − 5.28·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 7^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 7^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 3^{10} \cdot 7^{20} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(7.17757\times 10^{17}\)
Root analytic conductor: \(7.81265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 3^{10} \cdot 7^{20} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1084.765862\)
\(L(\frac12)\) \(\approx\) \(1084.765862\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T )^{10} \)
7 \( 1 \)
13 \( 1 - T + 23 T^{2} - 88 T^{3} + 244 T^{4} - 1806 T^{5} + 244 p T^{6} - 88 p^{2} T^{7} + 23 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 - 13 T^{2} + 53 T^{4} - 121 T^{6} + 1942 T^{8} - 16476 T^{10} + 1942 p^{2} T^{12} - 121 p^{4} T^{14} + 53 p^{6} T^{16} - 13 p^{8} T^{18} + p^{10} T^{20} \)
11 \( 1 - 28 T^{2} + 548 T^{4} - 8237 T^{6} + 102254 T^{8} - 1214300 T^{10} + 102254 p^{2} T^{12} - 8237 p^{4} T^{14} + 548 p^{6} T^{16} - 28 p^{8} T^{18} + p^{10} T^{20} \)
17 \( ( 1 + 2 T + 52 T^{2} + 33 T^{3} + 1166 T^{4} - 22 T^{5} + 1166 p T^{6} + 33 p^{2} T^{7} + 52 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 - 101 T^{2} + 4371 T^{4} - 101530 T^{6} + 3773 p^{2} T^{8} - 17032287 T^{10} + 3773 p^{4} T^{12} - 101530 p^{4} T^{14} + 4371 p^{6} T^{16} - 101 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 + 4 T + 76 T^{2} + 295 T^{3} + 3033 T^{4} + 8954 T^{5} + 3033 p T^{6} + 295 p^{2} T^{7} + 76 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( ( 1 + T + 10 T^{2} + 102 T^{3} + 560 T^{4} + 3634 T^{5} + 560 p T^{6} + 102 p^{2} T^{7} + 10 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 - 199 T^{2} + 20022 T^{4} - 1321316 T^{6} + 62802746 T^{8} - 2236428744 T^{10} + 62802746 p^{2} T^{12} - 1321316 p^{4} T^{14} + 20022 p^{6} T^{16} - 199 p^{8} T^{18} + p^{10} T^{20} \)
37 \( 1 - 272 T^{2} + 35058 T^{4} - 2861019 T^{6} + 165069438 T^{8} - 7052398464 T^{10} + 165069438 p^{2} T^{12} - 2861019 p^{4} T^{14} + 35058 p^{6} T^{16} - 272 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 98 T^{2} + 4554 T^{4} - 251867 T^{6} + 13268665 T^{8} - 545904606 T^{10} + 13268665 p^{2} T^{12} - 251867 p^{4} T^{14} + 4554 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} \)
43 \( ( 1 + 3 T + 184 T^{2} + 518 T^{3} + 14554 T^{4} + 33470 T^{5} + 14554 p T^{6} + 518 p^{2} T^{7} + 184 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 - 285 T^{2} + 40565 T^{4} - 3821356 T^{6} + 264293098 T^{8} - 14060400494 T^{10} + 264293098 p^{2} T^{12} - 3821356 p^{4} T^{14} + 40565 p^{6} T^{16} - 285 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 12 T + 72 T^{2} - 3 p T^{3} + 3640 T^{4} - 38322 T^{5} + 3640 p T^{6} - 3 p^{3} T^{7} + 72 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 - 328 T^{2} + 46790 T^{4} - 3885307 T^{6} + 226068472 T^{8} - 12326762100 T^{10} + 226068472 p^{2} T^{12} - 3885307 p^{4} T^{14} + 46790 p^{6} T^{16} - 328 p^{8} T^{18} + p^{10} T^{20} \)
61 \( ( 1 + 164 T^{2} + 109 T^{3} + 14260 T^{4} + 11840 T^{5} + 14260 p T^{6} + 109 p^{2} T^{7} + 164 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
67 \( 1 - 269 T^{2} + 37818 T^{4} - 3875556 T^{6} + 331645971 T^{8} - 24230245329 T^{10} + 331645971 p^{2} T^{12} - 3875556 p^{4} T^{14} + 37818 p^{6} T^{16} - 269 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 - 508 T^{2} + 121106 T^{4} - 18109091 T^{6} + 1923580424 T^{8} - 155049657368 T^{10} + 1923580424 p^{2} T^{12} - 18109091 p^{4} T^{14} + 121106 p^{6} T^{16} - 508 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 - 227 T^{2} + 35058 T^{4} - 4060008 T^{6} + 393782478 T^{8} - 31425811032 T^{10} + 393782478 p^{2} T^{12} - 4060008 p^{4} T^{14} + 35058 p^{6} T^{16} - 227 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 - 3 T + 212 T^{2} - 522 T^{3} + 24304 T^{4} - 58422 T^{5} + 24304 p T^{6} - 522 p^{2} T^{7} + 212 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 - 610 T^{2} + 175484 T^{4} - 31852831 T^{6} + 4089218707 T^{8} - 390599848590 T^{10} + 4089218707 p^{2} T^{12} - 31852831 p^{4} T^{14} + 175484 p^{6} T^{16} - 610 p^{8} T^{18} + p^{10} T^{20} \)
89 \( 1 - 9 p T^{2} + 295703 T^{4} - 66231973 T^{6} + 9964583800 T^{8} - 1052026924628 T^{10} + 9964583800 p^{2} T^{12} - 66231973 p^{4} T^{14} + 295703 p^{6} T^{16} - 9 p^{9} T^{18} + p^{10} T^{20} \)
97 \( 1 - 563 T^{2} + 123369 T^{4} - 11310265 T^{6} - 49803322 T^{8} + 79817466552 T^{10} - 49803322 p^{2} T^{12} - 11310265 p^{4} T^{14} + 123369 p^{6} T^{16} - 563 p^{8} T^{18} + p^{10} T^{20} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.67946763071070393654211026156, −2.60967745473657096871656290668, −2.58513853190481364436513088610, −2.51755992084137768441050044718, −2.22451705675180325531235143239, −2.14408458680273686535275049906, −2.11620387265160854158329001630, −2.05437133565137635844475990567, −2.02868832938374100836382376089, −1.92068487911993956843590453065, −1.85452646832937092606565010438, −1.83883563999883722537735616299, −1.81021712043676740808615071504, −1.46734295834235046679981103299, −1.43690299416533677429243350481, −1.22200176584574212814481860939, −1.18309819799261348522461099290, −1.15019819385375708104274462585, −1.13777073913877521264566573014, −0.944740672850397785054563922966, −0.59987057916423053006716175595, −0.55763163965513348349396218466, −0.49345184642283514331612404646, −0.37361568386646107184105382672, −0.32808769568800795428060910095, 0.32808769568800795428060910095, 0.37361568386646107184105382672, 0.49345184642283514331612404646, 0.55763163965513348349396218466, 0.59987057916423053006716175595, 0.944740672850397785054563922966, 1.13777073913877521264566573014, 1.15019819385375708104274462585, 1.18309819799261348522461099290, 1.22200176584574212814481860939, 1.43690299416533677429243350481, 1.46734295834235046679981103299, 1.81021712043676740808615071504, 1.83883563999883722537735616299, 1.85452646832937092606565010438, 1.92068487911993956843590453065, 2.02868832938374100836382376089, 2.05437133565137635844475990567, 2.11620387265160854158329001630, 2.14408458680273686535275049906, 2.22451705675180325531235143239, 2.51755992084137768441050044718, 2.58513853190481364436513088610, 2.60967745473657096871656290668, 2.67946763071070393654211026156

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

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