Dirichlet series
| L(s) = 1 | + 20·2-s − 3·3-s + 210·4-s − 10·5-s − 60·6-s − 20·7-s + 1.54e3·8-s − 15·9-s − 200·10-s − 17·11-s − 630·12-s − 23·13-s − 400·14-s + 30·15-s + 8.85e3·16-s − 21·17-s − 300·18-s − 22·19-s − 2.10e3·20-s + 60·21-s − 340·22-s + 15·23-s − 4.62e3·24-s − 460·26-s + 43·27-s − 4.20e3·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 14.1·2-s − 1.73·3-s + 105·4-s − 4.47·5-s − 24.4·6-s − 7.55·7-s + 544.·8-s − 5·9-s − 63.2·10-s − 5.12·11-s − 181.·12-s − 6.37·13-s − 106.·14-s + 7.74·15-s + 2.21e3·16-s − 5.09·17-s − 70.7·18-s − 5.04·19-s − 469.·20-s + 13.0·21-s − 72.4·22-s + 3.12·23-s − 943.·24-s − 90.2·26-s + 8.27·27-s − 793.·28-s − 0.557·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 431^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 431^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 7^{20} \cdot 431^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.54619\times 10^{33}\) |
| Root analytic conductor: | \(6.94130\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 2^{20} \cdot 7^{20} \cdot 431^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{20} \) |
| 7 | \( ( 1 + T )^{20} \) | |
| 431 | \( ( 1 - T )^{20} \) | |
| good | 3 | \( 1 + p T + 8 p T^{2} + 74 T^{3} + 113 p T^{4} + 962 T^{5} + 3383 T^{6} + 8780 T^{7} + 26099 T^{8} + 6916 p^{2} T^{9} + 163769 T^{10} + 362311 T^{11} + 865250 T^{12} + 596083 p T^{13} + 3943301 T^{14} + 7656907 T^{15} + 15762167 T^{16} + 28879544 T^{17} + 2070901 p^{3} T^{18} + 96915322 T^{19} + 177227896 T^{20} + 96915322 p T^{21} + 2070901 p^{5} T^{22} + 28879544 p^{3} T^{23} + 15762167 p^{4} T^{24} + 7656907 p^{5} T^{25} + 3943301 p^{6} T^{26} + 596083 p^{8} T^{27} + 865250 p^{8} T^{28} + 362311 p^{9} T^{29} + 163769 p^{10} T^{30} + 6916 p^{13} T^{31} + 26099 p^{12} T^{32} + 8780 p^{13} T^{33} + 3383 p^{14} T^{34} + 962 p^{15} T^{35} + 113 p^{17} T^{36} + 74 p^{17} T^{37} + 8 p^{19} T^{38} + p^{20} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + 2 p T + 4 p^{2} T^{2} + 673 T^{3} + 4184 T^{4} + 21792 T^{5} + 104853 T^{6} + 452522 T^{7} + 1822359 T^{8} + 6782363 T^{9} + 23788863 T^{10} + 78378628 T^{11} + 245415333 T^{12} + 729456057 T^{13} + 2074808653 T^{14} + 5644597267 T^{15} + 2955515449 p T^{16} + 37210029366 T^{17} + 90537843008 T^{18} + 212634700082 T^{19} + 483649864702 T^{20} + 212634700082 p T^{21} + 90537843008 p^{2} T^{22} + 37210029366 p^{3} T^{23} + 2955515449 p^{5} T^{24} + 5644597267 p^{5} T^{25} + 2074808653 p^{6} T^{26} + 729456057 p^{7} T^{27} + 245415333 p^{8} T^{28} + 78378628 p^{9} T^{29} + 23788863 p^{10} T^{30} + 6782363 p^{11} T^{31} + 1822359 p^{12} T^{32} + 452522 p^{13} T^{33} + 104853 p^{14} T^{34} + 21792 p^{15} T^{35} + 4184 p^{16} T^{36} + 673 p^{17} T^{37} + 4 p^{20} T^{38} + 2 p^{20} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 17 T + 262 T^{2} + 2779 T^{3} + 26652 T^{4} + 213525 T^{5} + 143032 p T^{6} + 10314287 T^{7} + 63054195 T^{8} + 353193376 T^{9} + 1864328516 T^{10} + 9171756147 T^{11} + 42880912543 T^{12} + 189038802409 T^{13} + 797906422256 T^{14} + 3204614953882 T^{15} + 12407669044248 T^{16} + 46052940406230 T^{17} + 165745632043125 T^{18} + 574888558062998 T^{19} + 1940309111902644 T^{20} + 574888558062998 p T^{21} + 165745632043125 p^{2} T^{22} + 46052940406230 p^{3} T^{23} + 12407669044248 p^{4} T^{24} + 3204614953882 p^{5} T^{25} + 797906422256 p^{6} T^{26} + 189038802409 p^{7} T^{27} + 42880912543 p^{8} T^{28} + 9171756147 p^{9} T^{29} + 1864328516 p^{10} T^{30} + 353193376 p^{11} T^{31} + 63054195 p^{12} T^{32} + 10314287 p^{13} T^{33} + 143032 p^{15} T^{34} + 213525 p^{15} T^{35} + 26652 p^{16} T^{36} + 2779 p^{17} T^{37} + 262 p^{18} T^{38} + 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 23 T + 392 T^{2} + 4926 T^{3} + 52532 T^{4} + 480757 T^{5} + 3934217 T^{6} + 29044664 T^{7} + 197015736 T^{8} + 1235565090 T^{9} + 7238080160 T^{10} + 39784666665 T^{11} + 206559196772 T^{12} + 1016381381784 T^{13} + 4762361875974 T^{14} + 21302343672547 T^{15} + 91285097803496 T^{16} + 375395005457828 T^{17} + 114239889574959 p T^{18} + 5656884167302232 T^{19} + 20774512439851354 T^{20} + 5656884167302232 p T^{21} + 114239889574959 p^{3} T^{22} + 375395005457828 p^{3} T^{23} + 91285097803496 p^{4} T^{24} + 21302343672547 p^{5} T^{25} + 4762361875974 p^{6} T^{26} + 1016381381784 p^{7} T^{27} + 206559196772 p^{8} T^{28} + 39784666665 p^{9} T^{29} + 7238080160 p^{10} T^{30} + 1235565090 p^{11} T^{31} + 197015736 p^{12} T^{32} + 29044664 p^{13} T^{33} + 3934217 p^{14} T^{34} + 480757 p^{15} T^{35} + 52532 p^{16} T^{36} + 4926 p^{17} T^{37} + 392 p^{18} T^{38} + 23 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 21 T + 415 T^{2} + 5504 T^{3} + 66910 T^{4} + 673732 T^{5} + 6279118 T^{6} + 52029319 T^{7} + 403459248 T^{8} + 2876014812 T^{9} + 19356717713 T^{10} + 122011298457 T^{11} + 730976601406 T^{12} + 4148161195383 T^{13} + 22481358998766 T^{14} + 116231956518111 T^{15} + 575788238408049 T^{16} + 2732903332778520 T^{17} + 12453611580210147 T^{18} + 54501867715981501 T^{19} + 229219753391983326 T^{20} + 54501867715981501 p T^{21} + 12453611580210147 p^{2} T^{22} + 2732903332778520 p^{3} T^{23} + 575788238408049 p^{4} T^{24} + 116231956518111 p^{5} T^{25} + 22481358998766 p^{6} T^{26} + 4148161195383 p^{7} T^{27} + 730976601406 p^{8} T^{28} + 122011298457 p^{9} T^{29} + 19356717713 p^{10} T^{30} + 2876014812 p^{11} T^{31} + 403459248 p^{12} T^{32} + 52029319 p^{13} T^{33} + 6279118 p^{14} T^{34} + 673732 p^{15} T^{35} + 66910 p^{16} T^{36} + 5504 p^{17} T^{37} + 415 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 22 T + 410 T^{2} + 5271 T^{3} + 60022 T^{4} + 568447 T^{5} + 4902807 T^{6} + 37414104 T^{7} + 264231175 T^{8} + 1698101660 T^{9} + 10189807352 T^{10} + 2969922875 p T^{11} + 292953307368 T^{12} + 1413134982330 T^{13} + 6397411260923 T^{14} + 27035266401000 T^{15} + 107683427946459 T^{16} + 407277069652428 T^{17} + 1503168432695800 T^{18} + 5684369830268923 T^{19} + 23355210441560646 T^{20} + 5684369830268923 p T^{21} + 1503168432695800 p^{2} T^{22} + 407277069652428 p^{3} T^{23} + 107683427946459 p^{4} T^{24} + 27035266401000 p^{5} T^{25} + 6397411260923 p^{6} T^{26} + 1413134982330 p^{7} T^{27} + 292953307368 p^{8} T^{28} + 2969922875 p^{10} T^{29} + 10189807352 p^{10} T^{30} + 1698101660 p^{11} T^{31} + 264231175 p^{12} T^{32} + 37414104 p^{13} T^{33} + 4902807 p^{14} T^{34} + 568447 p^{15} T^{35} + 60022 p^{16} T^{36} + 5271 p^{17} T^{37} + 410 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 15 T + 362 T^{2} - 4573 T^{3} + 64776 T^{4} - 694425 T^{5} + 7507590 T^{6} - 69633959 T^{7} + 629711091 T^{8} - 5150134823 T^{9} + 40635830348 T^{10} - 297498648538 T^{11} + 2095355167701 T^{12} - 13883837108274 T^{13} + 88485499271484 T^{14} - 534735606071258 T^{15} + 3109967920401641 T^{16} - 17228510967691727 T^{17} + 91905513586302764 T^{18} - 468104666251087544 T^{19} + 2296633639861040916 T^{20} - 468104666251087544 p T^{21} + 91905513586302764 p^{2} T^{22} - 17228510967691727 p^{3} T^{23} + 3109967920401641 p^{4} T^{24} - 534735606071258 p^{5} T^{25} + 88485499271484 p^{6} T^{26} - 13883837108274 p^{7} T^{27} + 2095355167701 p^{8} T^{28} - 297498648538 p^{9} T^{29} + 40635830348 p^{10} T^{30} - 5150134823 p^{11} T^{31} + 629711091 p^{12} T^{32} - 69633959 p^{13} T^{33} + 7507590 p^{14} T^{34} - 694425 p^{15} T^{35} + 64776 p^{16} T^{36} - 4573 p^{17} T^{37} + 362 p^{18} T^{38} - 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 3 T + 331 T^{2} + 423 T^{3} + 51662 T^{4} - 18520 T^{5} + 180349 p T^{6} - 9858076 T^{7} + 398403663 T^{8} - 1265816593 T^{9} + 24835338885 T^{10} - 101887175104 T^{11} + 1322294531857 T^{12} - 6113268305200 T^{13} + 61193236320988 T^{14} - 293936705461645 T^{15} + 2476236442769677 T^{16} - 11747320602378593 T^{17} + 87739751316611995 T^{18} - 398052468669020587 T^{19} + 2720647792635462696 T^{20} - 398052468669020587 p T^{21} + 87739751316611995 p^{2} T^{22} - 11747320602378593 p^{3} T^{23} + 2476236442769677 p^{4} T^{24} - 293936705461645 p^{5} T^{25} + 61193236320988 p^{6} T^{26} - 6113268305200 p^{7} T^{27} + 1322294531857 p^{8} T^{28} - 101887175104 p^{9} T^{29} + 24835338885 p^{10} T^{30} - 1265816593 p^{11} T^{31} + 398403663 p^{12} T^{32} - 9858076 p^{13} T^{33} + 180349 p^{15} T^{34} - 18520 p^{15} T^{35} + 51662 p^{16} T^{36} + 423 p^{17} T^{37} + 331 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 + 3 T + 318 T^{2} + 900 T^{3} + 50501 T^{4} + 141431 T^{5} + 5319189 T^{6} + 15364657 T^{7} + 13438698 p T^{8} + 1284572237 T^{9} + 25830096352 T^{10} + 87163735283 T^{11} + 1321652239222 T^{12} + 4942928994600 T^{13} + 57675670777759 T^{14} + 238319294150951 T^{15} + 2212397453382722 T^{16} + 9869949624379216 T^{17} + 2475983237897106 p T^{18} + 353178396508938568 T^{19} + 2463138350234883576 T^{20} + 353178396508938568 p T^{21} + 2475983237897106 p^{3} T^{22} + 9869949624379216 p^{3} T^{23} + 2212397453382722 p^{4} T^{24} + 238319294150951 p^{5} T^{25} + 57675670777759 p^{6} T^{26} + 4942928994600 p^{7} T^{27} + 1321652239222 p^{8} T^{28} + 87163735283 p^{9} T^{29} + 25830096352 p^{10} T^{30} + 1284572237 p^{11} T^{31} + 13438698 p^{13} T^{32} + 15364657 p^{13} T^{33} + 5319189 p^{14} T^{34} + 141431 p^{15} T^{35} + 50501 p^{16} T^{36} + 900 p^{17} T^{37} + 318 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + 14 T + 478 T^{2} + 5702 T^{3} + 109560 T^{4} + 1138869 T^{5} + 16076020 T^{6} + 148019276 T^{7} + 1700124271 T^{8} + 14039136258 T^{9} + 138411321670 T^{10} + 1035816187370 T^{11} + 245074916177 p T^{12} + 62130058954610 T^{13} + 495417304548222 T^{14} + 3144471431175427 T^{15} + 23344187851097695 T^{16} + 139195340121285919 T^{17} + 980454372544125171 T^{18} + 5572252652363538529 T^{19} + 37706149083875925300 T^{20} + 5572252652363538529 p T^{21} + 980454372544125171 p^{2} T^{22} + 139195340121285919 p^{3} T^{23} + 23344187851097695 p^{4} T^{24} + 3144471431175427 p^{5} T^{25} + 495417304548222 p^{6} T^{26} + 62130058954610 p^{7} T^{27} + 245074916177 p^{9} T^{28} + 1035816187370 p^{9} T^{29} + 138411321670 p^{10} T^{30} + 14039136258 p^{11} T^{31} + 1700124271 p^{12} T^{32} + 148019276 p^{13} T^{33} + 16076020 p^{14} T^{34} + 1138869 p^{15} T^{35} + 109560 p^{16} T^{36} + 5702 p^{17} T^{37} + 478 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 37 T + 1139 T^{2} + 24656 T^{3} + 472054 T^{4} + 7614375 T^{5} + 112223046 T^{6} + 1479800004 T^{7} + 18171684649 T^{8} + 205405186521 T^{9} + 2188376696738 T^{10} + 21805953366345 T^{11} + 206430025795442 T^{12} + 1845593687985130 T^{13} + 15759975623287710 T^{14} + 127879708450293802 T^{15} + 994537712033321889 T^{16} + 7377463235002134926 T^{17} + 52563825903578027799 T^{18} + \)\(35\!\cdots\!84\)\( T^{19} + \)\(23\!\cdots\!70\)\( T^{20} + \)\(35\!\cdots\!84\)\( p T^{21} + 52563825903578027799 p^{2} T^{22} + 7377463235002134926 p^{3} T^{23} + 994537712033321889 p^{4} T^{24} + 127879708450293802 p^{5} T^{25} + 15759975623287710 p^{6} T^{26} + 1845593687985130 p^{7} T^{27} + 206430025795442 p^{8} T^{28} + 21805953366345 p^{9} T^{29} + 2188376696738 p^{10} T^{30} + 205405186521 p^{11} T^{31} + 18171684649 p^{12} T^{32} + 1479800004 p^{13} T^{33} + 112223046 p^{14} T^{34} + 7614375 p^{15} T^{35} + 472054 p^{16} T^{36} + 24656 p^{17} T^{37} + 1139 p^{18} T^{38} + 37 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 5 T + 564 T^{2} + 3348 T^{3} + 157972 T^{4} + 24048 p T^{5} + 29342290 T^{6} + 200617849 T^{7} + 4051399264 T^{8} + 27842427634 T^{9} + 440846284132 T^{10} + 2967797910895 T^{11} + 39130680758781 T^{12} + 253757295317484 T^{13} + 2897921022280148 T^{14} + 17898474415146579 T^{15} + 181853735941488851 T^{16} + 1060583722015327077 T^{17} + 9770998761396000098 T^{18} + 53391621194146534697 T^{19} + \)\(45\!\cdots\!18\)\( T^{20} + 53391621194146534697 p T^{21} + 9770998761396000098 p^{2} T^{22} + 1060583722015327077 p^{3} T^{23} + 181853735941488851 p^{4} T^{24} + 17898474415146579 p^{5} T^{25} + 2897921022280148 p^{6} T^{26} + 253757295317484 p^{7} T^{27} + 39130680758781 p^{8} T^{28} + 2967797910895 p^{9} T^{29} + 440846284132 p^{10} T^{30} + 27842427634 p^{11} T^{31} + 4051399264 p^{12} T^{32} + 200617849 p^{13} T^{33} + 29342290 p^{14} T^{34} + 24048 p^{16} T^{35} + 157972 p^{16} T^{36} + 3348 p^{17} T^{37} + 564 p^{18} T^{38} + 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 + 29 T + 836 T^{2} + 16006 T^{3} + 294378 T^{4} + 4444483 T^{5} + 64396752 T^{6} + 823142364 T^{7} + 10129643860 T^{8} + 113744809482 T^{9} + 1233670062815 T^{10} + 12437434377662 T^{11} + 121440235879608 T^{12} + 1114487302876067 T^{13} + 9926053734937470 T^{14} + 83666217492160808 T^{15} + 685403554140201658 T^{16} + 5336323680174475005 T^{17} + 40417249015765775359 T^{18} + \)\(29\!\cdots\!74\)\( T^{19} + \)\(20\!\cdots\!58\)\( T^{20} + \)\(29\!\cdots\!74\)\( p T^{21} + 40417249015765775359 p^{2} T^{22} + 5336323680174475005 p^{3} T^{23} + 685403554140201658 p^{4} T^{24} + 83666217492160808 p^{5} T^{25} + 9926053734937470 p^{6} T^{26} + 1114487302876067 p^{7} T^{27} + 121440235879608 p^{8} T^{28} + 12437434377662 p^{9} T^{29} + 1233670062815 p^{10} T^{30} + 113744809482 p^{11} T^{31} + 10129643860 p^{12} T^{32} + 823142364 p^{13} T^{33} + 64396752 p^{14} T^{34} + 4444483 p^{15} T^{35} + 294378 p^{16} T^{36} + 16006 p^{17} T^{37} + 836 p^{18} T^{38} + 29 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 28 T + 855 T^{2} + 16971 T^{3} + 318453 T^{4} + 4989886 T^{5} + 72859181 T^{6} + 959522258 T^{7} + 11879169618 T^{8} + 136930261681 T^{9} + 1499641108359 T^{10} + 15540991233654 T^{11} + 154241728666272 T^{12} + 1462317158506270 T^{13} + 13348321475007699 T^{14} + 117074942188318834 T^{15} + 992078761734708850 T^{16} + 8105974828203638503 T^{17} + 64120597949792064174 T^{18} + \)\(48\!\cdots\!27\)\( T^{19} + \)\(36\!\cdots\!28\)\( T^{20} + \)\(48\!\cdots\!27\)\( p T^{21} + 64120597949792064174 p^{2} T^{22} + 8105974828203638503 p^{3} T^{23} + 992078761734708850 p^{4} T^{24} + 117074942188318834 p^{5} T^{25} + 13348321475007699 p^{6} T^{26} + 1462317158506270 p^{7} T^{27} + 154241728666272 p^{8} T^{28} + 15540991233654 p^{9} T^{29} + 1499641108359 p^{10} T^{30} + 136930261681 p^{11} T^{31} + 11879169618 p^{12} T^{32} + 959522258 p^{13} T^{33} + 72859181 p^{14} T^{34} + 4989886 p^{15} T^{35} + 318453 p^{16} T^{36} + 16971 p^{17} T^{37} + 855 p^{18} T^{38} + 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 47 T + 1788 T^{2} + 48119 T^{3} + 1126986 T^{4} + 22230123 T^{5} + 395860894 T^{6} + 6289733217 T^{7} + 92169409959 T^{8} + 1239099217883 T^{9} + 15580319162288 T^{10} + 182712600996040 T^{11} + 2024636845963401 T^{12} + 21162946437227494 T^{13} + 210740088201407780 T^{14} + 1996876129903102148 T^{15} + 18150432938574101217 T^{16} + \)\(15\!\cdots\!85\)\( T^{17} + \)\(13\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!04\)\( T^{19} + \)\(83\!\cdots\!72\)\( T^{20} + \)\(10\!\cdots\!04\)\( p T^{21} + \)\(13\!\cdots\!02\)\( p^{2} T^{22} + \)\(15\!\cdots\!85\)\( p^{3} T^{23} + 18150432938574101217 p^{4} T^{24} + 1996876129903102148 p^{5} T^{25} + 210740088201407780 p^{6} T^{26} + 21162946437227494 p^{7} T^{27} + 2024636845963401 p^{8} T^{28} + 182712600996040 p^{9} T^{29} + 15580319162288 p^{10} T^{30} + 1239099217883 p^{11} T^{31} + 92169409959 p^{12} T^{32} + 6289733217 p^{13} T^{33} + 395860894 p^{14} T^{34} + 22230123 p^{15} T^{35} + 1126986 p^{16} T^{36} + 48119 p^{17} T^{37} + 1788 p^{18} T^{38} + 47 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 13 T + 543 T^{2} + 5537 T^{3} + 135356 T^{4} + 1158259 T^{5} + 22151239 T^{6} + 169961620 T^{7} + 2842190167 T^{8} + 20535982161 T^{9} + 311621710393 T^{10} + 2156274759569 T^{11} + 30008264843466 T^{12} + 198638743651611 T^{13} + 2566312885568540 T^{14} + 16252555189553676 T^{15} + 197668959333327174 T^{16} + 1201249528469311764 T^{17} + 13861604702787399114 T^{18} + 80762667542124740506 T^{19} + \)\(88\!\cdots\!22\)\( T^{20} + 80762667542124740506 p T^{21} + 13861604702787399114 p^{2} T^{22} + 1201249528469311764 p^{3} T^{23} + 197668959333327174 p^{4} T^{24} + 16252555189553676 p^{5} T^{25} + 2566312885568540 p^{6} T^{26} + 198638743651611 p^{7} T^{27} + 30008264843466 p^{8} T^{28} + 2156274759569 p^{9} T^{29} + 311621710393 p^{10} T^{30} + 20535982161 p^{11} T^{31} + 2842190167 p^{12} T^{32} + 169961620 p^{13} T^{33} + 22151239 p^{14} T^{34} + 1158259 p^{15} T^{35} + 135356 p^{16} T^{36} + 5537 p^{17} T^{37} + 543 p^{18} T^{38} + 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 24 T + 655 T^{2} + 11807 T^{3} + 206811 T^{4} + 3136202 T^{5} + 44832552 T^{6} + 596798322 T^{7} + 7501071068 T^{8} + 89816005294 T^{9} + 1026691105841 T^{10} + 11272421156478 T^{11} + 119262652414856 T^{12} + 1216266583348386 T^{13} + 12022878639293925 T^{14} + 114938154792189479 T^{15} + 1068172138954193764 T^{16} + 9630693245059776611 T^{17} + 84464162833322292609 T^{18} + \)\(72\!\cdots\!87\)\( T^{19} + \)\(59\!\cdots\!32\)\( T^{20} + \)\(72\!\cdots\!87\)\( p T^{21} + 84464162833322292609 p^{2} T^{22} + 9630693245059776611 p^{3} T^{23} + 1068172138954193764 p^{4} T^{24} + 114938154792189479 p^{5} T^{25} + 12022878639293925 p^{6} T^{26} + 1216266583348386 p^{7} T^{27} + 119262652414856 p^{8} T^{28} + 11272421156478 p^{9} T^{29} + 1026691105841 p^{10} T^{30} + 89816005294 p^{11} T^{31} + 7501071068 p^{12} T^{32} + 596798322 p^{13} T^{33} + 44832552 p^{14} T^{34} + 3136202 p^{15} T^{35} + 206811 p^{16} T^{36} + 11807 p^{17} T^{37} + 655 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 22 T + 913 T^{2} + 15614 T^{3} + 375610 T^{4} + 5418429 T^{5} + 97821779 T^{6} + 1250498050 T^{7} + 18689775413 T^{8} + 218083949274 T^{9} + 2834871504001 T^{10} + 30720985748034 T^{11} + 357651564487925 T^{12} + 3637271856489393 T^{13} + 38684748866261100 T^{14} + 371541335571880405 T^{15} + 3661669132324374044 T^{16} + 33317878341909397282 T^{17} + \)\(30\!\cdots\!56\)\( T^{18} + \)\(26\!\cdots\!21\)\( T^{19} + \)\(23\!\cdots\!52\)\( T^{20} + \)\(26\!\cdots\!21\)\( p T^{21} + \)\(30\!\cdots\!56\)\( p^{2} T^{22} + 33317878341909397282 p^{3} T^{23} + 3661669132324374044 p^{4} T^{24} + 371541335571880405 p^{5} T^{25} + 38684748866261100 p^{6} T^{26} + 3637271856489393 p^{7} T^{27} + 357651564487925 p^{8} T^{28} + 30720985748034 p^{9} T^{29} + 2834871504001 p^{10} T^{30} + 218083949274 p^{11} T^{31} + 18689775413 p^{12} T^{32} + 1250498050 p^{13} T^{33} + 97821779 p^{14} T^{34} + 5418429 p^{15} T^{35} + 375610 p^{16} T^{36} + 15614 p^{17} T^{37} + 913 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 37 T + 1322 T^{2} + 31457 T^{3} + 707479 T^{4} + 13098092 T^{5} + 3161415 p T^{6} + 3598777065 T^{7} + 53811554660 T^{8} + 736820848086 T^{9} + 9729356612710 T^{10} + 119864595167890 T^{11} + 1429796735217682 T^{12} + 16090479386235061 T^{13} + 175808744649259140 T^{14} + 1824721474066883767 T^{15} + 18421784068159602034 T^{16} + \)\(17\!\cdots\!61\)\( T^{17} + \)\(16\!\cdots\!17\)\( T^{18} + \)\(14\!\cdots\!96\)\( T^{19} + \)\(13\!\cdots\!40\)\( T^{20} + \)\(14\!\cdots\!96\)\( p T^{21} + \)\(16\!\cdots\!17\)\( p^{2} T^{22} + \)\(17\!\cdots\!61\)\( p^{3} T^{23} + 18421784068159602034 p^{4} T^{24} + 1824721474066883767 p^{5} T^{25} + 175808744649259140 p^{6} T^{26} + 16090479386235061 p^{7} T^{27} + 1429796735217682 p^{8} T^{28} + 119864595167890 p^{9} T^{29} + 9729356612710 p^{10} T^{30} + 736820848086 p^{11} T^{31} + 53811554660 p^{12} T^{32} + 3598777065 p^{13} T^{33} + 3161415 p^{15} T^{34} + 13098092 p^{15} T^{35} + 707479 p^{16} T^{36} + 31457 p^{17} T^{37} + 1322 p^{18} T^{38} + 37 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 25 T + 1250 T^{2} - 24578 T^{3} + 714684 T^{4} - 11827289 T^{5} + 257795206 T^{6} - 3730126203 T^{7} + 67020308433 T^{8} - 867930014509 T^{9} + 13487302851081 T^{10} - 158728718389989 T^{11} + 2194288625379001 T^{12} - 23710528353777159 T^{13} + 296877642502245672 T^{14} - 2965745503247346096 T^{15} + 34035469574124251962 T^{16} - \)\(31\!\cdots\!87\)\( T^{17} + \)\(33\!\cdots\!83\)\( T^{18} - \)\(28\!\cdots\!15\)\( T^{19} + \)\(28\!\cdots\!22\)\( T^{20} - \)\(28\!\cdots\!15\)\( p T^{21} + \)\(33\!\cdots\!83\)\( p^{2} T^{22} - \)\(31\!\cdots\!87\)\( p^{3} T^{23} + 34035469574124251962 p^{4} T^{24} - 2965745503247346096 p^{5} T^{25} + 296877642502245672 p^{6} T^{26} - 23710528353777159 p^{7} T^{27} + 2194288625379001 p^{8} T^{28} - 158728718389989 p^{9} T^{29} + 13487302851081 p^{10} T^{30} - 867930014509 p^{11} T^{31} + 67020308433 p^{12} T^{32} - 3730126203 p^{13} T^{33} + 257795206 p^{14} T^{34} - 11827289 p^{15} T^{35} + 714684 p^{16} T^{36} - 24578 p^{17} T^{37} + 1250 p^{18} T^{38} - 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 33 T + 1134 T^{2} + 25077 T^{3} + 538999 T^{4} + 9350713 T^{5} + 157289285 T^{6} + 2310569800 T^{7} + 33131147213 T^{8} + 431195103972 T^{9} + 5516658793839 T^{10} + 65485766141129 T^{11} + 768275389211482 T^{12} + 8473190596979554 T^{13} + 92684268230633733 T^{14} + 960676708836587901 T^{15} + 9899744401622803947 T^{16} + 97159557348164041570 T^{17} + \)\(94\!\cdots\!95\)\( T^{18} + \)\(88\!\cdots\!99\)\( T^{19} + \)\(82\!\cdots\!80\)\( T^{20} + \)\(88\!\cdots\!99\)\( p T^{21} + \)\(94\!\cdots\!95\)\( p^{2} T^{22} + 97159557348164041570 p^{3} T^{23} + 9899744401622803947 p^{4} T^{24} + 960676708836587901 p^{5} T^{25} + 92684268230633733 p^{6} T^{26} + 8473190596979554 p^{7} T^{27} + 768275389211482 p^{8} T^{28} + 65485766141129 p^{9} T^{29} + 5516658793839 p^{10} T^{30} + 431195103972 p^{11} T^{31} + 33131147213 p^{12} T^{32} + 2310569800 p^{13} T^{33} + 157289285 p^{14} T^{34} + 9350713 p^{15} T^{35} + 538999 p^{16} T^{36} + 25077 p^{17} T^{37} + 1134 p^{18} T^{38} + 33 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 71 T + 3343 T^{2} + 117202 T^{3} + 3412879 T^{4} + 85408826 T^{5} + 1899135453 T^{6} + 38133185174 T^{7} + 702129680973 T^{8} + 11963638122922 T^{9} + 190258595791697 T^{10} + 31908305894677 p T^{11} + 39991336529210688 T^{12} + 533225123678772761 T^{13} + 6753936578406729148 T^{14} + 81449969478068436467 T^{15} + \)\(93\!\cdots\!31\)\( T^{16} + \)\(10\!\cdots\!51\)\( T^{17} + \)\(10\!\cdots\!14\)\( T^{18} + \)\(10\!\cdots\!93\)\( T^{19} + \)\(10\!\cdots\!78\)\( T^{20} + \)\(10\!\cdots\!93\)\( p T^{21} + \)\(10\!\cdots\!14\)\( p^{2} T^{22} + \)\(10\!\cdots\!51\)\( p^{3} T^{23} + \)\(93\!\cdots\!31\)\( p^{4} T^{24} + 81449969478068436467 p^{5} T^{25} + 6753936578406729148 p^{6} T^{26} + 533225123678772761 p^{7} T^{27} + 39991336529210688 p^{8} T^{28} + 31908305894677 p^{10} T^{29} + 190258595791697 p^{10} T^{30} + 11963638122922 p^{11} T^{31} + 702129680973 p^{12} T^{32} + 38133185174 p^{13} T^{33} + 1899135453 p^{14} T^{34} + 85408826 p^{15} T^{35} + 3412879 p^{16} T^{36} + 117202 p^{17} T^{37} + 3343 p^{18} T^{38} + 71 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 51 T + 2293 T^{2} + 71265 T^{3} + 2008195 T^{4} + 47361843 T^{5} + 1035451775 T^{6} + 20193675075 T^{7} + 370369666479 T^{8} + 6244076099934 T^{9} + 99953768435771 T^{10} + 1495502248233244 T^{11} + 21385023953653722 T^{12} + 288724751613226927 T^{13} + 3742241751744269966 T^{14} + 46088072796316313612 T^{15} + \)\(54\!\cdots\!93\)\( T^{16} + \)\(61\!\cdots\!58\)\( T^{17} + \)\(67\!\cdots\!19\)\( T^{18} + \)\(70\!\cdots\!07\)\( T^{19} + \)\(70\!\cdots\!68\)\( T^{20} + \)\(70\!\cdots\!07\)\( p T^{21} + \)\(67\!\cdots\!19\)\( p^{2} T^{22} + \)\(61\!\cdots\!58\)\( p^{3} T^{23} + \)\(54\!\cdots\!93\)\( p^{4} T^{24} + 46088072796316313612 p^{5} T^{25} + 3742241751744269966 p^{6} T^{26} + 288724751613226927 p^{7} T^{27} + 21385023953653722 p^{8} T^{28} + 1495502248233244 p^{9} T^{29} + 99953768435771 p^{10} T^{30} + 6244076099934 p^{11} T^{31} + 370369666479 p^{12} T^{32} + 20193675075 p^{13} T^{33} + 1035451775 p^{14} T^{34} + 47361843 p^{15} T^{35} + 2008195 p^{16} T^{36} + 71265 p^{17} T^{37} + 2293 p^{18} T^{38} + 51 p^{19} T^{39} + p^{20} T^{40} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.39644170349299151156363289254, −2.32553108887746632524351007834, −2.30936128563608608535096585443, −2.27964863830811446150941135765, −2.26810809488564176859041501207, −1.97007063237055858358159218743, −1.95971874208126337994353477697, −1.88583338079477005463203106120, −1.83913968253974551236907263612, −1.79307957351389850226315758614, −1.72805721448238320512537101080, −1.69089229085634375186461964289, −1.69020459461500700772399437520, −1.67984489850429807150607063146, −1.59131687563656190987979085415, −1.57211115348708598016323971345, −1.42114552542714497617010832004, −1.38811946671560530328694244619, −1.34800836362455330124231726181, −1.28838303786589172314287973223, −1.22605592256699952433387709721, −1.13260192532082706645205190370, −1.00247040969992397494363531129, −0.951426305797181985413685011165, −0.899696229146440621722693272920, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.899696229146440621722693272920, 0.951426305797181985413685011165, 1.00247040969992397494363531129, 1.13260192532082706645205190370, 1.22605592256699952433387709721, 1.28838303786589172314287973223, 1.34800836362455330124231726181, 1.38811946671560530328694244619, 1.42114552542714497617010832004, 1.57211115348708598016323971345, 1.59131687563656190987979085415, 1.67984489850429807150607063146, 1.69020459461500700772399437520, 1.69089229085634375186461964289, 1.72805721448238320512537101080, 1.79307957351389850226315758614, 1.83913968253974551236907263612, 1.88583338079477005463203106120, 1.95971874208126337994353477697, 1.97007063237055858358159218743, 2.26810809488564176859041501207, 2.27964863830811446150941135765, 2.30936128563608608535096585443, 2.32553108887746632524351007834, 2.39644170349299151156363289254
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.