Properties

Degree $20$
Conductor $1.037\times 10^{27}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $10$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 8·3-s − 5-s + 32·6-s − 5·7-s + 23·8-s + 16·9-s + 4·10-s − 3·11-s − 18·13-s + 20·14-s + 8·15-s − 31·16-s − 11·17-s − 64·18-s + 40·21-s + 12·22-s − 2·23-s − 184·24-s − 38·25-s + 72·26-s + 50·27-s − 9·29-s − 32·30-s − 22·31-s − 40·32-s + 24·33-s + ⋯
L(s)  = 1  − 2.82·2-s − 4.61·3-s − 0.447·5-s + 13.0·6-s − 1.88·7-s + 8.13·8-s + 16/3·9-s + 1.26·10-s − 0.904·11-s − 4.99·13-s + 5.34·14-s + 2.06·15-s − 7.75·16-s − 2.66·17-s − 15.0·18-s + 8.72·21-s + 2.55·22-s − 0.417·23-s − 37.5·24-s − 7.59·25-s + 14.1·26-s + 9.62·27-s − 1.67·29-s − 5.84·30-s − 3.95·31-s − 7.07·32-s + 4.17·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(503^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(503^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(503^{10}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{503} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(10\)
Selberg data: \((20,\ 503^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( ( 1 + T )^{10} \)
good2 \( 1 + p^{2} T + p^{4} T^{2} + 41 T^{3} + 103 T^{4} + 13 p^{4} T^{5} + 103 p^{2} T^{6} + 707 T^{7} + 595 p T^{8} + 897 p T^{9} + 2665 T^{10} + 897 p^{2} T^{11} + 595 p^{3} T^{12} + 707 p^{3} T^{13} + 103 p^{6} T^{14} + 13 p^{9} T^{15} + 103 p^{6} T^{16} + 41 p^{7} T^{17} + p^{12} T^{18} + p^{11} T^{19} + p^{10} T^{20} \)
3 \( 1 + 8 T + 16 p T^{2} + 206 T^{3} + 752 T^{4} + 769 p T^{5} + 700 p^{2} T^{6} + 15215 T^{7} + 33404 T^{8} + 66208 T^{9} + 120343 T^{10} + 66208 p T^{11} + 33404 p^{2} T^{12} + 15215 p^{3} T^{13} + 700 p^{6} T^{14} + 769 p^{6} T^{15} + 752 p^{6} T^{16} + 206 p^{7} T^{17} + 16 p^{9} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
5 \( 1 + T + 39 T^{2} + 38 T^{3} + 726 T^{4} + 662 T^{5} + 8471 T^{6} + 7016 T^{7} + 68463 T^{8} + 50106 T^{9} + 400081 T^{10} + 50106 p T^{11} + 68463 p^{2} T^{12} + 7016 p^{3} T^{13} + 8471 p^{4} T^{14} + 662 p^{5} T^{15} + 726 p^{6} T^{16} + 38 p^{7} T^{17} + 39 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
7 \( 1 + 5 T + 61 T^{2} + 251 T^{3} + 1709 T^{4} + 5912 T^{5} + 29171 T^{6} + 85970 T^{7} + 337948 T^{8} + 851532 T^{9} + 2785579 T^{10} + 851532 p T^{11} + 337948 p^{2} T^{12} + 85970 p^{3} T^{13} + 29171 p^{4} T^{14} + 5912 p^{5} T^{15} + 1709 p^{6} T^{16} + 251 p^{7} T^{17} + 61 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + 3 T + 69 T^{2} + 217 T^{3} + 2352 T^{4} + 7445 T^{5} + 52841 T^{6} + 159886 T^{7} + 869048 T^{8} + 2400415 T^{9} + 10894513 T^{10} + 2400415 p T^{11} + 869048 p^{2} T^{12} + 159886 p^{3} T^{13} + 52841 p^{4} T^{14} + 7445 p^{5} T^{15} + 2352 p^{6} T^{16} + 217 p^{7} T^{17} + 69 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 + 18 T + 250 T^{2} + 2413 T^{3} + 19873 T^{4} + 134672 T^{5} + 810062 T^{6} + 325417 p T^{7} + 19976351 T^{8} + 83675331 T^{9} + 319368199 T^{10} + 83675331 p T^{11} + 19976351 p^{2} T^{12} + 325417 p^{4} T^{13} + 810062 p^{4} T^{14} + 134672 p^{5} T^{15} + 19873 p^{6} T^{16} + 2413 p^{7} T^{17} + 250 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 + 11 T + 172 T^{2} + 1316 T^{3} + 12110 T^{4} + 72481 T^{5} + 497736 T^{6} + 2460829 T^{7} + 13750697 T^{8} + 57744215 T^{9} + 272963791 T^{10} + 57744215 p T^{11} + 13750697 p^{2} T^{12} + 2460829 p^{3} T^{13} + 497736 p^{4} T^{14} + 72481 p^{5} T^{15} + 12110 p^{6} T^{16} + 1316 p^{7} T^{17} + 172 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 117 T^{2} + 144 T^{3} + 6545 T^{4} + 14584 T^{5} + 240551 T^{6} + 681477 T^{7} + 6591561 T^{8} + 19407794 T^{9} + 141051753 T^{10} + 19407794 p T^{11} + 6591561 p^{2} T^{12} + 681477 p^{3} T^{13} + 240551 p^{4} T^{14} + 14584 p^{5} T^{15} + 6545 p^{6} T^{16} + 144 p^{7} T^{17} + 117 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 + 2 T + 128 T^{2} + 139 T^{3} + 7117 T^{4} - 755 T^{5} + 224173 T^{6} - 407616 T^{7} + 4690668 T^{8} - 19318174 T^{9} + 93011881 T^{10} - 19318174 p T^{11} + 4690668 p^{2} T^{12} - 407616 p^{3} T^{13} + 224173 p^{4} T^{14} - 755 p^{5} T^{15} + 7117 p^{6} T^{16} + 139 p^{7} T^{17} + 128 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 + 9 T + 225 T^{2} + 1647 T^{3} + 23252 T^{4} + 143611 T^{5} + 1501329 T^{6} + 8016660 T^{7} + 68262730 T^{8} + 317447419 T^{9} + 2290933469 T^{10} + 317447419 p T^{11} + 68262730 p^{2} T^{12} + 8016660 p^{3} T^{13} + 1501329 p^{4} T^{14} + 143611 p^{5} T^{15} + 23252 p^{6} T^{16} + 1647 p^{7} T^{17} + 225 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + 22 T + 437 T^{2} + 5905 T^{3} + 71196 T^{4} + 707620 T^{5} + 6360319 T^{6} + 49929045 T^{7} + 358058017 T^{8} + 2291835411 T^{9} + 13467173163 T^{10} + 2291835411 p T^{11} + 358058017 p^{2} T^{12} + 49929045 p^{3} T^{13} + 6360319 p^{4} T^{14} + 707620 p^{5} T^{15} + 71196 p^{6} T^{16} + 5905 p^{7} T^{17} + 437 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 35 T + 807 T^{2} + 13514 T^{3} + 185245 T^{4} + 2135413 T^{5} + 21450226 T^{6} + 189788329 T^{7} + 1500972503 T^{8} + 10649984634 T^{9} + 1843721455 p T^{10} + 10649984634 p T^{11} + 1500972503 p^{2} T^{12} + 189788329 p^{3} T^{13} + 21450226 p^{4} T^{14} + 2135413 p^{5} T^{15} + 185245 p^{6} T^{16} + 13514 p^{7} T^{17} + 807 p^{8} T^{18} + 35 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 4 T + 260 T^{2} + 1417 T^{3} + 32755 T^{4} + 218127 T^{5} + 2678340 T^{6} + 482650 p T^{7} + 159907986 T^{8} + 1182434807 T^{9} + 7385189807 T^{10} + 1182434807 p T^{11} + 159907986 p^{2} T^{12} + 482650 p^{4} T^{13} + 2678340 p^{4} T^{14} + 218127 p^{5} T^{15} + 32755 p^{6} T^{16} + 1417 p^{7} T^{17} + 260 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 20 T + 435 T^{2} + 5094 T^{3} + 60693 T^{4} + 456684 T^{5} + 3529418 T^{6} + 14396035 T^{7} + 65477767 T^{8} - 145261781 T^{9} - 450150713 T^{10} - 145261781 p T^{11} + 65477767 p^{2} T^{12} + 14396035 p^{3} T^{13} + 3529418 p^{4} T^{14} + 456684 p^{5} T^{15} + 60693 p^{6} T^{16} + 5094 p^{7} T^{17} + 435 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - 7 T + 322 T^{2} - 2204 T^{3} + 50553 T^{4} - 325252 T^{5} + 5105346 T^{6} - 30078097 T^{7} + 367899299 T^{8} - 1942791244 T^{9} + 19840453349 T^{10} - 1942791244 p T^{11} + 367899299 p^{2} T^{12} - 30078097 p^{3} T^{13} + 5105346 p^{4} T^{14} - 325252 p^{5} T^{15} + 50553 p^{6} T^{16} - 2204 p^{7} T^{17} + 322 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 + 24 T + 570 T^{2} + 8461 T^{3} + 123312 T^{4} + 1397686 T^{5} + 294465 p T^{6} + 145668337 T^{7} + 1341977182 T^{8} + 10633420505 T^{9} + 83159919093 T^{10} + 10633420505 p T^{11} + 1341977182 p^{2} T^{12} + 145668337 p^{3} T^{13} + 294465 p^{5} T^{14} + 1397686 p^{5} T^{15} + 123312 p^{6} T^{16} + 8461 p^{7} T^{17} + 570 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 17 T + 467 T^{2} - 5784 T^{3} + 84135 T^{4} - 790560 T^{5} + 7647144 T^{6} - 56238779 T^{7} + 400513752 T^{8} - 2655972904 T^{9} + 18692560589 T^{10} - 2655972904 p T^{11} + 400513752 p^{2} T^{12} - 56238779 p^{3} T^{13} + 7647144 p^{4} T^{14} - 790560 p^{5} T^{15} + 84135 p^{6} T^{16} - 5784 p^{7} T^{17} + 467 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 4 T + 252 T^{2} + 859 T^{3} + 35750 T^{4} + 141968 T^{5} + 3623379 T^{6} + 15927450 T^{7} + 286566685 T^{8} + 1331963236 T^{9} + 19096375159 T^{10} + 1331963236 p T^{11} + 286566685 p^{2} T^{12} + 15927450 p^{3} T^{13} + 3623379 p^{4} T^{14} + 141968 p^{5} T^{15} + 35750 p^{6} T^{16} + 859 p^{7} T^{17} + 252 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 6 T + 318 T^{2} + 2402 T^{3} + 61276 T^{4} + 443521 T^{5} + 8166786 T^{6} + 55393313 T^{7} + 807832572 T^{8} + 4959723492 T^{9} + 61619242847 T^{10} + 4959723492 p T^{11} + 807832572 p^{2} T^{12} + 55393313 p^{3} T^{13} + 8166786 p^{4} T^{14} + 443521 p^{5} T^{15} + 61276 p^{6} T^{16} + 2402 p^{7} T^{17} + 318 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + T + 262 T^{2} + 1233 T^{3} + 39245 T^{4} + 293739 T^{5} + 4405465 T^{6} + 40998321 T^{7} + 397241676 T^{8} + 3994326482 T^{9} + 30391775917 T^{10} + 3994326482 p T^{11} + 397241676 p^{2} T^{12} + 40998321 p^{3} T^{13} + 4405465 p^{4} T^{14} + 293739 p^{5} T^{15} + 39245 p^{6} T^{16} + 1233 p^{7} T^{17} + 262 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 + 31 T + 857 T^{2} + 16625 T^{3} + 280173 T^{4} + 4020157 T^{5} + 51396432 T^{6} + 589991993 T^{7} + 6192642640 T^{8} + 59433187850 T^{9} + 529794375619 T^{10} + 59433187850 p T^{11} + 6192642640 p^{2} T^{12} + 589991993 p^{3} T^{13} + 51396432 p^{4} T^{14} + 4020157 p^{5} T^{15} + 280173 p^{6} T^{16} + 16625 p^{7} T^{17} + 857 p^{8} T^{18} + 31 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 10 T + 248 T^{2} + 2937 T^{3} + 29662 T^{4} + 273591 T^{5} + 1835094 T^{6} + 1901304 T^{7} - 54353511 T^{8} - 1412662644 T^{9} - 15400667207 T^{10} - 1412662644 p T^{11} - 54353511 p^{2} T^{12} + 1901304 p^{3} T^{13} + 1835094 p^{4} T^{14} + 273591 p^{5} T^{15} + 29662 p^{6} T^{16} + 2937 p^{7} T^{17} + 248 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 22 T + 669 T^{2} - 9824 T^{3} + 169480 T^{4} - 1798297 T^{5} + 22307432 T^{6} - 175385507 T^{7} + 1816492439 T^{8} - 11655093474 T^{9} + 132417751709 T^{10} - 11655093474 p T^{11} + 1816492439 p^{2} T^{12} - 175385507 p^{3} T^{13} + 22307432 p^{4} T^{14} - 1798297 p^{5} T^{15} + 169480 p^{6} T^{16} - 9824 p^{7} T^{17} + 669 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - T + 510 T^{2} - 62 T^{3} + 113775 T^{4} + 104240 T^{5} + 14626103 T^{6} + 33572815 T^{7} + 1283471757 T^{8} + 5086267491 T^{9} + 104337188501 T^{10} + 5086267491 p T^{11} + 1283471757 p^{2} T^{12} + 33572815 p^{3} T^{13} + 14626103 p^{4} T^{14} + 104240 p^{5} T^{15} + 113775 p^{6} T^{16} - 62 p^{7} T^{17} + 510 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 57 T + 1999 T^{2} + 49485 T^{3} + 981253 T^{4} + 16180839 T^{5} + 233385980 T^{6} + 3001658219 T^{7} + 35423192688 T^{8} + 386010871872 T^{9} + 3937832201969 T^{10} + 386010871872 p T^{11} + 35423192688 p^{2} T^{12} + 3001658219 p^{3} T^{13} + 233385980 p^{4} T^{14} + 16180839 p^{5} T^{15} + 981253 p^{6} T^{16} + 49485 p^{7} T^{17} + 1999 p^{8} T^{18} + 57 p^{9} T^{19} + p^{10} T^{20} \)
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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

−4.85735929312955023060199259222, −4.81882432559567811103129986471, −4.69238975365214526165116832123, −4.51376071483096784551376712580, −4.43493388878886222689102822589, −4.08316204472350628528746160400, −4.02222312527322453501192208442, −3.85280977327434200316008754894, −3.71132593037249286610528280034, −3.69557217551358433867422098067, −3.63041197872983686430071714236, −3.47317555755971725644831080250, −3.29163548459967138294514493509, −3.12983915588039610610705239119, −3.04430776434398889730145649122, −2.95781778542714237960065650684, −2.54896388395685297495362863307, −2.38032565242628041145185462801, −2.29259548583577659881946130124, −2.14541617555194133492562465260, −2.07683284652257114995983312896, −1.94242025813742524563653306179, −1.52148486580040692606158735436, −1.49228221554600116989213035098, −1.47409392221607039132248049202, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.47409392221607039132248049202, 1.49228221554600116989213035098, 1.52148486580040692606158735436, 1.94242025813742524563653306179, 2.07683284652257114995983312896, 2.14541617555194133492562465260, 2.29259548583577659881946130124, 2.38032565242628041145185462801, 2.54896388395685297495362863307, 2.95781778542714237960065650684, 3.04430776434398889730145649122, 3.12983915588039610610705239119, 3.29163548459967138294514493509, 3.47317555755971725644831080250, 3.63041197872983686430071714236, 3.69557217551358433867422098067, 3.71132593037249286610528280034, 3.85280977327434200316008754894, 4.02222312527322453501192208442, 4.08316204472350628528746160400, 4.43493388878886222689102822589, 4.51376071483096784551376712580, 4.69238975365214526165116832123, 4.81882432559567811103129986471, 4.85735929312955023060199259222

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

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