Properties

Degree 20
Conductor $ 2^{40} \cdot 503^{10} $
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  + 8·3-s − 5-s + 5·7-s + 16·9-s + 3·11-s − 18·13-s − 8·15-s − 11·17-s + 40·21-s + 2·23-s − 38·25-s − 50·27-s − 9·29-s + 22·31-s + 24·33-s − 5·35-s − 35·37-s − 144·39-s − 4·41-s + 20·43-s − 16·45-s − 7·47-s − 36·49-s − 88·51-s − 24·53-s − 3·55-s − 17·59-s + ⋯
L(s)  = 1  + 4.61·3-s − 0.447·5-s + 1.88·7-s + 16/3·9-s + 0.904·11-s − 4.99·13-s − 2.06·15-s − 2.66·17-s + 8.72·21-s + 0.417·23-s − 7.59·25-s − 9.62·27-s − 1.67·29-s + 3.95·31-s + 4.17·33-s − 0.845·35-s − 5.75·37-s − 23.0·39-s − 0.624·41-s + 3.04·43-s − 2.38·45-s − 1.02·47-s − 5.14·49-s − 12.3·51-s − 3.29·53-s − 0.404·55-s − 2.21·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40} \cdot 503^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40} \cdot 503^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{40} \cdot 503^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  10
Selberg data  =  $(20,\ 2^{40} \cdot 503^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \(F_p\) is a polynomial of degree 20. If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 \( 1 \)
503 \( ( 1 - T )^{10} \)
good3 \( 1 - 8 T + 16 p T^{2} - 206 T^{3} + 752 T^{4} - 769 p T^{5} + 700 p^{2} T^{6} - 15215 T^{7} + 33404 T^{8} - 66208 T^{9} + 120343 T^{10} - 66208 p T^{11} + 33404 p^{2} T^{12} - 15215 p^{3} T^{13} + 700 p^{6} T^{14} - 769 p^{6} T^{15} + 752 p^{6} T^{16} - 206 p^{7} T^{17} + 16 p^{9} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.09704685421057847738343832529, −3.06429794135468887247100271352, −3.02665180558042759351881926450, −2.84943288730317361237745264838, −2.71136164798044342617013109858, −2.55190422313249098132454860680, −2.51012258927727108144791064871, −2.36857156174397122903587858360, −2.31255365548548628294672534861, −2.28081791294447833927469708487, −2.23379838047914627923596684579, −2.18338612849222785981301334443, −2.16522293425519912462191933341, −2.12452960313394461885542522535, −2.11173797827183793348824636030, −1.89193302942981337190482801975, −1.68386886340034345648761537884, −1.58510729611366036278814425202, −1.47456512985157684613966121565, −1.41546261645210980502326801104, −1.41187947184039973695053820193, −1.37955790277592905801675394156, −1.25729578586778742184228759951, −1.05648492980120982710552705863, −0.831289083494322303812140328840, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.831289083494322303812140328840, 1.05648492980120982710552705863, 1.25729578586778742184228759951, 1.37955790277592905801675394156, 1.41187947184039973695053820193, 1.41546261645210980502326801104, 1.47456512985157684613966121565, 1.58510729611366036278814425202, 1.68386886340034345648761537884, 1.89193302942981337190482801975, 2.11173797827183793348824636030, 2.12452960313394461885542522535, 2.16522293425519912462191933341, 2.18338612849222785981301334443, 2.23379838047914627923596684579, 2.28081791294447833927469708487, 2.31255365548548628294672534861, 2.36857156174397122903587858360, 2.51012258927727108144791064871, 2.55190422313249098132454860680, 2.71136164798044342617013109858, 2.84943288730317361237745264838, 3.02665180558042759351881926450, 3.06429794135468887247100271352, 3.09704685421057847738343832529

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

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