Properties

Label 20-546e10-1.1-c1e10-0-1
Degree $20$
Conductor $2.355\times 10^{27}$
Sign $1$
Analytic cond. $2.48140\times 10^{6}$
Root an. cond. $2.08802$
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  + 5·2-s − 10·3-s + 10·4-s − 2·5-s − 50·6-s + 4·7-s + 5·8-s + 55·9-s − 10·10-s − 12·11-s − 100·12-s − 4·13-s + 20·14-s + 20·15-s − 20·16-s + 4·17-s + 275·18-s − 6·19-s − 20·20-s − 40·21-s − 60·22-s + 6·23-s − 50·24-s + 14·25-s − 20·26-s − 220·27-s + 40·28-s + ⋯
L(s)  = 1  + 3.53·2-s − 5.77·3-s + 5·4-s − 0.894·5-s − 20.4·6-s + 1.51·7-s + 1.76·8-s + 55/3·9-s − 3.16·10-s − 3.61·11-s − 28.8·12-s − 1.10·13-s + 5.34·14-s + 5.16·15-s − 5·16-s + 0.970·17-s + 64.8·18-s − 1.37·19-s − 4.47·20-s − 8.72·21-s − 12.7·22-s + 1.25·23-s − 10.2·24-s + 14/5·25-s − 3.92·26-s − 42.3·27-s + 7.55·28-s + ⋯

Functional equation

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

Invariants

Degree: \(20\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(2.48140\times 10^{6}\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.07160233784\)
\(L(\frac12)\) \(\approx\) \(0.07160233784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{5} \)
3 \( ( 1 + T )^{10} \)
7 \( 1 - 4 T + 18 T^{2} - 26 T^{3} + 62 T^{4} + 3 T^{5} + 62 p T^{6} - 26 p^{2} T^{7} + 18 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13 \( 1 + 4 T + 24 T^{2} + 44 T^{3} + 86 T^{4} + 15 T^{5} + 86 p T^{6} + 44 p^{2} T^{7} + 24 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 + 2 T - 2 p T^{2} - 44 T^{3} + 4 p T^{4} + 344 T^{5} + 368 T^{6} - 1484 T^{7} - 766 p T^{8} + 2784 T^{9} + 22431 T^{10} + 2784 p T^{11} - 766 p^{3} T^{12} - 1484 p^{3} T^{13} + 368 p^{4} T^{14} + 344 p^{5} T^{15} + 4 p^{7} T^{16} - 44 p^{7} T^{17} - 2 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
11 \( ( 1 + 6 T + 43 T^{2} + 199 T^{3} + 793 T^{4} + 2956 T^{5} + 793 p T^{6} + 199 p^{2} T^{7} + 43 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 4 T - 49 T^{2} + 142 T^{3} + 1568 T^{4} - 2791 T^{5} - 35242 T^{6} + 34954 T^{7} + 616861 T^{8} - 8985 p T^{9} - 10586085 T^{10} - 8985 p^{2} T^{11} + 616861 p^{2} T^{12} + 34954 p^{3} T^{13} - 35242 p^{4} T^{14} - 2791 p^{5} T^{15} + 1568 p^{6} T^{16} + 142 p^{7} T^{17} - 49 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
19 \( ( 1 + 3 T + 26 T^{2} + 119 T^{3} + 778 T^{4} + 2245 T^{5} + 778 p T^{6} + 119 p^{2} T^{7} + 26 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 - 6 T - 7 T^{2} - 228 T^{3} + 1986 T^{4} + 1929 T^{5} + 31332 T^{6} - 428832 T^{7} - 72459 T^{8} - 1481475 T^{9} + 55247559 T^{10} - 1481475 p T^{11} - 72459 p^{2} T^{12} - 428832 p^{3} T^{13} + 31332 p^{4} T^{14} + 1929 p^{5} T^{15} + 1986 p^{6} T^{16} - 228 p^{7} T^{17} - 7 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 85 T^{2} - 54 T^{3} + 3630 T^{4} + 3375 T^{5} - 96336 T^{6} - 63828 T^{7} + 1960125 T^{8} + 182925 T^{9} - 42333513 T^{10} + 182925 p T^{11} + 1960125 p^{2} T^{12} - 63828 p^{3} T^{13} - 96336 p^{4} T^{14} + 3375 p^{5} T^{15} + 3630 p^{6} T^{16} - 54 p^{7} T^{17} - 85 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 + 10 T - 29 T^{2} - 192 T^{3} + 3510 T^{4} + 6771 T^{5} - 90246 T^{6} + 119556 T^{7} + 1279461 T^{8} - 4544141 T^{9} - 16306697 T^{10} - 4544141 p T^{11} + 1279461 p^{2} T^{12} + 119556 p^{3} T^{13} - 90246 p^{4} T^{14} + 6771 p^{5} T^{15} + 3510 p^{6} T^{16} - 192 p^{7} T^{17} - 29 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - T - 146 T^{2} + 183 T^{3} + 11604 T^{4} - 13482 T^{5} - 700002 T^{6} + 405093 T^{7} + 34841964 T^{8} - 4401244 T^{9} - 1422580340 T^{10} - 4401244 p T^{11} + 34841964 p^{2} T^{12} + 405093 p^{3} T^{13} - 700002 p^{4} T^{14} - 13482 p^{5} T^{15} + 11604 p^{6} T^{16} + 183 p^{7} T^{17} - 146 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 4 T - 157 T^{2} - 366 T^{3} + 14718 T^{4} + 17565 T^{5} - 1035036 T^{6} - 695484 T^{7} + 57489417 T^{8} + 14323171 T^{9} - 2587515041 T^{10} + 14323171 p T^{11} + 57489417 p^{2} T^{12} - 695484 p^{3} T^{13} - 1035036 p^{4} T^{14} + 17565 p^{5} T^{15} + 14718 p^{6} T^{16} - 366 p^{7} T^{17} - 157 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 3 T - 104 T^{2} + 223 T^{3} + 6210 T^{4} - 11084 T^{5} - 116252 T^{6} + 216573 T^{7} - 4797452 T^{8} - 4910440 T^{9} + 487444146 T^{10} - 4910440 p T^{11} - 4797452 p^{2} T^{12} + 216573 p^{3} T^{13} - 116252 p^{4} T^{14} - 11084 p^{5} T^{15} + 6210 p^{6} T^{16} + 223 p^{7} T^{17} - 104 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 15 T + 44 T^{2} - 685 T^{3} - 8256 T^{4} - 47792 T^{5} - 110249 T^{6} + 1247169 T^{7} + 15610607 T^{8} + 42731021 T^{9} - 121164885 T^{10} + 42731021 p T^{11} + 15610607 p^{2} T^{12} + 1247169 p^{3} T^{13} - 110249 p^{4} T^{14} - 47792 p^{5} T^{15} - 8256 p^{6} T^{16} - 685 p^{7} T^{17} + 44 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 + 17 T + 14 T^{2} - 497 T^{3} + 6152 T^{4} + 22370 T^{5} - 660061 T^{6} - 2005769 T^{7} + 23181553 T^{8} - 11517771 T^{9} - 1562966193 T^{10} - 11517771 p T^{11} + 23181553 p^{2} T^{12} - 2005769 p^{3} T^{13} - 660061 p^{4} T^{14} + 22370 p^{5} T^{15} + 6152 p^{6} T^{16} - 497 p^{7} T^{17} + 14 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 2 T - 124 T^{2} + 1568 T^{3} + 7442 T^{4} - 183224 T^{5} + 570644 T^{6} + 13803548 T^{7} - 103282556 T^{8} - 320380188 T^{9} + 9099276549 T^{10} - 320380188 p T^{11} - 103282556 p^{2} T^{12} + 13803548 p^{3} T^{13} + 570644 p^{4} T^{14} - 183224 p^{5} T^{15} + 7442 p^{6} T^{16} + 1568 p^{7} T^{17} - 124 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
61 \( ( 1 + 11 T + 297 T^{2} + 2284 T^{3} + 34730 T^{4} + 196866 T^{5} + 34730 p T^{6} + 2284 p^{2} T^{7} + 297 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( ( 1 - T + 198 T^{2} - 69 T^{3} + 18072 T^{4} - 288 T^{5} + 18072 p T^{6} - 69 p^{2} T^{7} + 198 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( 1 - 18 T - 40 T^{2} + 1108 T^{3} + 16884 T^{4} - 76762 T^{5} - 2223296 T^{6} + 6435546 T^{7} + 147203672 T^{8} - 9125804 T^{9} - 12149042091 T^{10} - 9125804 p T^{11} + 147203672 p^{2} T^{12} + 6435546 p^{3} T^{13} - 2223296 p^{4} T^{14} - 76762 p^{5} T^{15} + 16884 p^{6} T^{16} + 1108 p^{7} T^{17} - 40 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 12 T + 43 T^{2} - 472 T^{3} - 87 p T^{4} + 132407 T^{5} - 768992 T^{6} + 7360740 T^{7} + 20692501 T^{8} - 799259342 T^{9} + 4955096646 T^{10} - 799259342 p T^{11} + 20692501 p^{2} T^{12} + 7360740 p^{3} T^{13} - 768992 p^{4} T^{14} + 132407 p^{5} T^{15} - 87 p^{7} T^{16} - 472 p^{7} T^{17} + 43 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 4 T - 308 T^{2} - 1152 T^{3} + 53712 T^{4} + 168192 T^{5} - 6769152 T^{6} - 13333176 T^{7} + 690029892 T^{8} + 445430536 T^{9} - 59061818879 T^{10} + 445430536 p T^{11} + 690029892 p^{2} T^{12} - 13333176 p^{3} T^{13} - 6769152 p^{4} T^{14} + 168192 p^{5} T^{15} + 53712 p^{6} T^{16} - 1152 p^{7} T^{17} - 308 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
83 \( ( 1 + 355 T^{2} - 27 T^{3} + 54355 T^{4} - 3996 T^{5} + 54355 p T^{6} - 27 p^{2} T^{7} + 355 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
89 \( 1 - 7 T + 68 T^{2} - 2423 T^{3} + 3410 T^{4} - 56596 T^{5} + 1990889 T^{6} + 9620437 T^{7} + 63224731 T^{8} - 1239058803 T^{9} - 10986048021 T^{10} - 1239058803 p T^{11} + 63224731 p^{2} T^{12} + 9620437 p^{3} T^{13} + 1990889 p^{4} T^{14} - 56596 p^{5} T^{15} + 3410 p^{6} T^{16} - 2423 p^{7} T^{17} + 68 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 6 T - 206 T^{2} - 2336 T^{3} + 22716 T^{4} + 420385 T^{5} + 876934 T^{6} - 45385950 T^{7} - 466860704 T^{8} + 1843252112 T^{9} + 68681302440 T^{10} + 1843252112 p T^{11} - 466860704 p^{2} T^{12} - 45385950 p^{3} T^{13} + 876934 p^{4} T^{14} + 420385 p^{5} T^{15} + 22716 p^{6} T^{16} - 2336 p^{7} T^{17} - 206 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
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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

−3.99029162104211404013029941581, −3.96594555080335639466954088774, −3.96229893822566388107521417481, −3.89889002283052094607472285230, −3.83606540649433245123022017920, −3.70643146179463328657720337125, −3.41859210599648905175481375319, −3.35456553261346161861713284635, −3.21644988148118895484457962904, −3.17429964897042730785043956508, −2.75146418175952106417266364340, −2.72881980641518619846922454648, −2.68557859451419441553069165072, −2.52623540578326875931186483580, −2.30036888111904558541440380672, −2.22507047565512331569853501570, −1.92230899134626669703283414713, −1.77558295864066770436406442467, −1.38029416162187881624212788115, −1.26052798868033874402035704239, −1.19594232437629699262268737683, −1.18423878233764156723338362058, −0.46159957991740011945290513467, −0.24349707769701556348661131074, −0.13482871418623752304333742908, 0.13482871418623752304333742908, 0.24349707769701556348661131074, 0.46159957991740011945290513467, 1.18423878233764156723338362058, 1.19594232437629699262268737683, 1.26052798868033874402035704239, 1.38029416162187881624212788115, 1.77558295864066770436406442467, 1.92230899134626669703283414713, 2.22507047565512331569853501570, 2.30036888111904558541440380672, 2.52623540578326875931186483580, 2.68557859451419441553069165072, 2.72881980641518619846922454648, 2.75146418175952106417266364340, 3.17429964897042730785043956508, 3.21644988148118895484457962904, 3.35456553261346161861713284635, 3.41859210599648905175481375319, 3.70643146179463328657720337125, 3.83606540649433245123022017920, 3.89889002283052094607472285230, 3.96229893822566388107521417481, 3.96594555080335639466954088774, 3.99029162104211404013029941581

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

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