Properties

Label 20-4005e10-1.1-c1e10-0-0
Degree $20$
Conductor $1.062\times 10^{36}$
Sign $1$
Analytic cond. $1.11891\times 10^{15}$
Root an. cond. $5.65509$
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  − 6·2-s + 15·4-s − 10·5-s + 7·7-s − 19·8-s + 60·10-s + 10·11-s + 7·13-s − 42·14-s + 9·16-s − 11·17-s + 10·19-s − 150·20-s − 60·22-s − 6·23-s + 55·25-s − 42·26-s + 105·28-s + 3·29-s + 12·31-s + 12·32-s + 66·34-s − 70·35-s − 19·37-s − 60·38-s + 190·40-s − 13·41-s + ⋯
L(s)  = 1  − 4.24·2-s + 15/2·4-s − 4.47·5-s + 2.64·7-s − 6.71·8-s + 18.9·10-s + 3.01·11-s + 1.94·13-s − 11.2·14-s + 9/4·16-s − 2.66·17-s + 2.29·19-s − 33.5·20-s − 12.7·22-s − 1.25·23-s + 11·25-s − 8.23·26-s + 19.8·28-s + 0.557·29-s + 2.15·31-s + 2.12·32-s + 11.3·34-s − 11.8·35-s − 3.12·37-s − 9.73·38-s + 30.0·40-s − 2.03·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{20} \cdot 5^{10} \cdot 89^{10}\)
Sign: $1$
Analytic conductor: \(1.11891\times 10^{15}\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{20} \cdot 5^{10} \cdot 89^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1136931227\)
\(L(\frac12)\) \(\approx\) \(0.1136931227\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + T )^{10} \)
89 \( ( 1 + T )^{10} \)
good2 \( 1 + 3 p T + 21 T^{2} + 55 T^{3} + 15 p^{3} T^{4} + 57 p^{2} T^{5} + 197 p T^{6} + 629 T^{7} + 471 p T^{8} + 1353 T^{9} + 1919 T^{10} + 1353 p T^{11} + 471 p^{3} T^{12} + 629 p^{3} T^{13} + 197 p^{5} T^{14} + 57 p^{7} T^{15} + 15 p^{9} T^{16} + 55 p^{7} T^{17} + 21 p^{8} T^{18} + 3 p^{10} T^{19} + p^{10} T^{20} \)
7 \( 1 - p T + 33 T^{2} - 95 T^{3} + 41 p T^{4} - 857 T^{5} + 3287 T^{6} - 10168 T^{7} + 29210 T^{8} - 9188 p T^{9} + 23708 p T^{10} - 9188 p^{2} T^{11} + 29210 p^{2} T^{12} - 10168 p^{3} T^{13} + 3287 p^{4} T^{14} - 857 p^{5} T^{15} + 41 p^{7} T^{16} - 95 p^{7} T^{17} + 33 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \)
11 \( 1 - 10 T + 74 T^{2} - 450 T^{3} + 2273 T^{4} - 10244 T^{5} + 42712 T^{6} - 163028 T^{7} + 592982 T^{8} - 2062960 T^{9} + 6918588 T^{10} - 2062960 p T^{11} + 592982 p^{2} T^{12} - 163028 p^{3} T^{13} + 42712 p^{4} T^{14} - 10244 p^{5} T^{15} + 2273 p^{6} T^{16} - 450 p^{7} T^{17} + 74 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 - 7 T + 5 p T^{2} - 29 p T^{3} + 2415 T^{4} - 12141 T^{5} + 61969 T^{6} - 270814 T^{7} + 1175276 T^{8} - 4567824 T^{9} + 17348254 T^{10} - 4567824 p T^{11} + 1175276 p^{2} T^{12} - 270814 p^{3} T^{13} + 61969 p^{4} T^{14} - 12141 p^{5} T^{15} + 2415 p^{6} T^{16} - 29 p^{8} T^{17} + 5 p^{9} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 + 11 T + 165 T^{2} + 1281 T^{3} + 11321 T^{4} + 69739 T^{5} + 460979 T^{6} + 2370332 T^{7} + 12737098 T^{8} + 55835562 T^{9} + 253110738 T^{10} + 55835562 p T^{11} + 12737098 p^{2} T^{12} + 2370332 p^{3} T^{13} + 460979 p^{4} T^{14} + 69739 p^{5} T^{15} + 11321 p^{6} T^{16} + 1281 p^{7} T^{17} + 165 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 10 T + 112 T^{2} - 710 T^{3} + 4853 T^{4} - 23472 T^{5} + 121320 T^{6} - 484240 T^{7} + 2139282 T^{8} - 7953396 T^{9} + 36233872 T^{10} - 7953396 p T^{11} + 2139282 p^{2} T^{12} - 484240 p^{3} T^{13} + 121320 p^{4} T^{14} - 23472 p^{5} T^{15} + 4853 p^{6} T^{16} - 710 p^{7} T^{17} + 112 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 6 T + 110 T^{2} + 658 T^{3} + 6153 T^{4} + 34192 T^{5} + 222920 T^{6} + 1187424 T^{7} + 264874 p T^{8} + 31930956 T^{9} + 145556756 T^{10} + 31930956 p T^{11} + 264874 p^{3} T^{12} + 1187424 p^{3} T^{13} + 222920 p^{4} T^{14} + 34192 p^{5} T^{15} + 6153 p^{6} T^{16} + 658 p^{7} T^{17} + 110 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 3 T + 161 T^{2} - 285 T^{3} + 12667 T^{4} - 12087 T^{5} + 675355 T^{6} - 298668 T^{7} + 27269546 T^{8} - 4748708 T^{9} + 876949202 T^{10} - 4748708 p T^{11} + 27269546 p^{2} T^{12} - 298668 p^{3} T^{13} + 675355 p^{4} T^{14} - 12087 p^{5} T^{15} + 12667 p^{6} T^{16} - 285 p^{7} T^{17} + 161 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 12 T + 170 T^{2} - 1700 T^{3} + 15033 T^{4} - 122184 T^{5} + 905336 T^{6} - 6227336 T^{7} + 40893926 T^{8} - 247582168 T^{9} + 1434456060 T^{10} - 247582168 p T^{11} + 40893926 p^{2} T^{12} - 6227336 p^{3} T^{13} + 905336 p^{4} T^{14} - 122184 p^{5} T^{15} + 15033 p^{6} T^{16} - 1700 p^{7} T^{17} + 170 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 19 T + 425 T^{2} + 5455 T^{3} + 72231 T^{4} + 710795 T^{5} + 6998309 T^{6} + 55928102 T^{7} + 443294366 T^{8} + 2956166396 T^{9} + 19505505806 T^{10} + 2956166396 p T^{11} + 443294366 p^{2} T^{12} + 55928102 p^{3} T^{13} + 6998309 p^{4} T^{14} + 710795 p^{5} T^{15} + 72231 p^{6} T^{16} + 5455 p^{7} T^{17} + 425 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 13 T + 279 T^{2} + 2517 T^{3} + 32589 T^{4} + 221909 T^{5} + 2241499 T^{6} + 12200824 T^{7} + 110628762 T^{8} + 521806424 T^{9} + 4681667490 T^{10} + 521806424 p T^{11} + 110628762 p^{2} T^{12} + 12200824 p^{3} T^{13} + 2241499 p^{4} T^{14} + 221909 p^{5} T^{15} + 32589 p^{6} T^{16} + 2517 p^{7} T^{17} + 279 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 9 T + 269 T^{2} - 1899 T^{3} + 35173 T^{4} - 206861 T^{5} + 2991603 T^{6} - 15030816 T^{7} + 185972412 T^{8} - 818475042 T^{9} + 8987075000 T^{10} - 818475042 p T^{11} + 185972412 p^{2} T^{12} - 15030816 p^{3} T^{13} + 2991603 p^{4} T^{14} - 206861 p^{5} T^{15} + 35173 p^{6} T^{16} - 1899 p^{7} T^{17} + 269 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 21 T + 471 T^{2} + 6465 T^{3} + 88083 T^{4} + 925123 T^{5} + 9554121 T^{6} + 82516602 T^{7} + 703326422 T^{8} + 5196452810 T^{9} + 38053483252 T^{10} + 5196452810 p T^{11} + 703326422 p^{2} T^{12} + 82516602 p^{3} T^{13} + 9554121 p^{4} T^{14} + 925123 p^{5} T^{15} + 88083 p^{6} T^{16} + 6465 p^{7} T^{17} + 471 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 + 7 T + 247 T^{2} + 1277 T^{3} + 32449 T^{4} + 138541 T^{5} + 3052655 T^{6} + 11254832 T^{7} + 223022992 T^{8} + 721437508 T^{9} + 13062964726 T^{10} + 721437508 p T^{11} + 223022992 p^{2} T^{12} + 11254832 p^{3} T^{13} + 3052655 p^{4} T^{14} + 138541 p^{5} T^{15} + 32449 p^{6} T^{16} + 1277 p^{7} T^{17} + 247 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 19 T + 509 T^{2} - 7069 T^{3} + 110421 T^{4} - 1243387 T^{5} + 14546465 T^{6} - 139836842 T^{7} + 1341017824 T^{8} - 11197843906 T^{9} + 91418000172 T^{10} - 11197843906 p T^{11} + 1341017824 p^{2} T^{12} - 139836842 p^{3} T^{13} + 14546465 p^{4} T^{14} - 1243387 p^{5} T^{15} + 110421 p^{6} T^{16} - 7069 p^{7} T^{17} + 509 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 4 T + 386 T^{2} - 868 T^{3} + 70937 T^{4} - 77344 T^{5} + 8583064 T^{6} - 3824464 T^{7} + 769666446 T^{8} - 138361432 T^{9} + 53227597324 T^{10} - 138361432 p T^{11} + 769666446 p^{2} T^{12} - 3824464 p^{3} T^{13} + 8583064 p^{4} T^{14} - 77344 p^{5} T^{15} + 70937 p^{6} T^{16} - 868 p^{7} T^{17} + 386 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 6 T + 350 T^{2} + 2662 T^{3} + 63785 T^{4} + 500324 T^{5} + 8102056 T^{6} + 58775668 T^{7} + 771341406 T^{8} + 5098209400 T^{9} + 57582032244 T^{10} + 5098209400 p T^{11} + 771341406 p^{2} T^{12} + 58775668 p^{3} T^{13} + 8102056 p^{4} T^{14} + 500324 p^{5} T^{15} + 63785 p^{6} T^{16} + 2662 p^{7} T^{17} + 350 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 6 T + 408 T^{2} - 1310 T^{3} + 1139 p T^{4} - 139092 T^{5} + 11047544 T^{6} - 10652532 T^{7} + 1140578986 T^{8} - 714273992 T^{9} + 91516654112 T^{10} - 714273992 p T^{11} + 1140578986 p^{2} T^{12} - 10652532 p^{3} T^{13} + 11047544 p^{4} T^{14} - 139092 p^{5} T^{15} + 1139 p^{7} T^{16} - 1310 p^{7} T^{17} + 408 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 6 T + 296 T^{2} - 2194 T^{3} + 49213 T^{4} - 374068 T^{5} + 6311560 T^{6} - 43740652 T^{7} + 629824978 T^{8} - 4038801784 T^{9} + 50310309792 T^{10} - 4038801784 p T^{11} + 629824978 p^{2} T^{12} - 43740652 p^{3} T^{13} + 6311560 p^{4} T^{14} - 374068 p^{5} T^{15} + 49213 p^{6} T^{16} - 2194 p^{7} T^{17} + 296 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 25 T + 841 T^{2} - 15215 T^{3} + 293767 T^{4} - 4188385 T^{5} + 59159975 T^{6} - 695380616 T^{7} + 7854823128 T^{8} - 77854031824 T^{9} + 732944496240 T^{10} - 77854031824 p T^{11} + 7854823128 p^{2} T^{12} - 695380616 p^{3} T^{13} + 59159975 p^{4} T^{14} - 4188385 p^{5} T^{15} + 293767 p^{6} T^{16} - 15215 p^{7} T^{17} + 841 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 22 T + 676 T^{2} - 10862 T^{3} + 195989 T^{4} - 2528044 T^{5} + 34612952 T^{6} - 378409340 T^{7} + 4326317010 T^{8} - 41211193360 T^{9} + 409114238728 T^{10} - 41211193360 p T^{11} + 4326317010 p^{2} T^{12} - 378409340 p^{3} T^{13} + 34612952 p^{4} T^{14} - 2528044 p^{5} T^{15} + 195989 p^{6} T^{16} - 10862 p^{7} T^{17} + 676 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 40 T + 1118 T^{2} - 23504 T^{3} + 433833 T^{4} - 6960336 T^{5} + 101141816 T^{6} - 1322290384 T^{7} + 15958027446 T^{8} - 176598350248 T^{9} + 1814489192980 T^{10} - 176598350248 p T^{11} + 15958027446 p^{2} T^{12} - 1322290384 p^{3} T^{13} + 101141816 p^{4} T^{14} - 6960336 p^{5} T^{15} + 433833 p^{6} T^{16} - 23504 p^{7} T^{17} + 1118 p^{8} T^{18} - 40 p^{9} T^{19} + p^{10} T^{20} \)
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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

−2.87243507553485964811502793117, −2.82652706373219810003356124236, −2.77302619131822395066670409969, −2.53434035366216295219600508393, −2.44116435677163941887114056506, −2.35407224318714214050555601698, −2.11582142338449808243424394004, −2.11228131043783259105226673125, −1.81111289837489359428468094327, −1.79632366163846854585363410023, −1.76390876279286718710611181392, −1.74691502941393225586124596072, −1.66233410362521498748820508937, −1.60409485023675361194119536887, −1.37429709636709085135492848410, −1.29020890862314286542309330414, −0.947957122264042433291601338131, −0.921360554786814639113905243402, −0.871275117835323473901073013734, −0.71900720786793808576667928665, −0.59470281854348861264286521860, −0.55976044168691071630771921353, −0.53470603913670208307755366861, −0.41567282958089703379617941586, −0.07057302653986205757876577513, 0.07057302653986205757876577513, 0.41567282958089703379617941586, 0.53470603913670208307755366861, 0.55976044168691071630771921353, 0.59470281854348861264286521860, 0.71900720786793808576667928665, 0.871275117835323473901073013734, 0.921360554786814639113905243402, 0.947957122264042433291601338131, 1.29020890862314286542309330414, 1.37429709636709085135492848410, 1.60409485023675361194119536887, 1.66233410362521498748820508937, 1.74691502941393225586124596072, 1.76390876279286718710611181392, 1.79632366163846854585363410023, 1.81111289837489359428468094327, 2.11228131043783259105226673125, 2.11582142338449808243424394004, 2.35407224318714214050555601698, 2.44116435677163941887114056506, 2.53434035366216295219600508393, 2.77302619131822395066670409969, 2.82652706373219810003356124236, 2.87243507553485964811502793117

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

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