Dirichlet series
| L(s) = 1 | + 10·3-s + 6·5-s + 105·7-s − 70·9-s + 62·11-s + 148·13-s + 60·15-s + 132·17-s + 260·19-s + 1.05e3·21-s − 26·23-s − 693·25-s − 972·27-s + 12·29-s + 144·31-s + 620·33-s + 630·35-s + 274·37-s + 1.48e3·39-s − 615·41-s + 986·43-s − 420·45-s + 44·47-s + 5.88e3·49-s + 1.32e3·51-s − 366·53-s + 372·55-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.536·5-s + 5.66·7-s − 2.59·9-s + 1.69·11-s + 3.15·13-s + 1.03·15-s + 1.88·17-s + 3.13·19-s + 10.9·21-s − 0.235·23-s − 5.54·25-s − 6.92·27-s + 0.0768·29-s + 0.834·31-s + 3.27·33-s + 3.04·35-s + 1.21·37-s + 6.07·39-s − 2.34·41-s + 3.49·43-s − 1.39·45-s + 0.136·47-s + 17.1·49-s + 3.62·51-s − 0.948·53-s + 0.912·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{15} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{15} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{30} \cdot 7^{15} \cdot 41^{15}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.89814\times 10^{27}\) |
| Root analytic conductor: | \(8.23007\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((30,\ 2^{30} \cdot 7^{15} \cdot 41^{15} ,\ ( \ : [3/2]^{15} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(7044.863663\) |
| \(L(\frac12)\) | \(\approx\) | \(7044.863663\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - p T )^{15} \) | |
| 41 | \( ( 1 + p T )^{15} \) | |
| good | 3 | \( 1 - 10 T + 170 T^{2} - 476 p T^{3} + 14854 T^{4} - 108422 T^{5} + 916163 T^{6} - 2027074 p T^{7} + 15058942 p T^{8} - 278314916 T^{9} + 1867759781 T^{10} - 10712074672 T^{11} + 7352344880 p^{2} T^{12} - 355012268198 T^{13} + 2038714390771 T^{14} - 10258797303376 T^{15} + 2038714390771 p^{3} T^{16} - 355012268198 p^{6} T^{17} + 7352344880 p^{11} T^{18} - 10712074672 p^{12} T^{19} + 1867759781 p^{15} T^{20} - 278314916 p^{18} T^{21} + 15058942 p^{22} T^{22} - 2027074 p^{25} T^{23} + 916163 p^{27} T^{24} - 108422 p^{30} T^{25} + 14854 p^{33} T^{26} - 476 p^{37} T^{27} + 170 p^{39} T^{28} - 10 p^{42} T^{29} + p^{45} T^{30} \) |
| 5 | \( 1 - 6 T + 729 T^{2} - 6066 T^{3} + 296451 T^{4} - 555134 p T^{5} + 89731728 T^{6} - 168705842 p T^{7} + 21711591328 T^{8} - 198133654142 T^{9} + 4325487385151 T^{10} - 7599782626342 p T^{11} + 729488356855669 T^{12} - 6070540340479974 T^{13} + 105813275645448811 T^{14} - 821164561492351788 T^{15} + 105813275645448811 p^{3} T^{16} - 6070540340479974 p^{6} T^{17} + 729488356855669 p^{9} T^{18} - 7599782626342 p^{13} T^{19} + 4325487385151 p^{15} T^{20} - 198133654142 p^{18} T^{21} + 21711591328 p^{21} T^{22} - 168705842 p^{25} T^{23} + 89731728 p^{27} T^{24} - 555134 p^{31} T^{25} + 296451 p^{33} T^{26} - 6066 p^{36} T^{27} + 729 p^{39} T^{28} - 6 p^{42} T^{29} + p^{45} T^{30} \) | |
| 11 | \( 1 - 62 T + 9693 T^{2} - 353060 T^{3} + 37920506 T^{4} - 890859104 T^{5} + 103542045132 T^{6} - 1881615778408 T^{7} + 247104798829846 T^{8} - 3688624202139500 T^{9} + 489560790263567776 T^{10} - 5631437161022373980 T^{11} + \)\(82\!\cdots\!27\)\( T^{12} - \)\(82\!\cdots\!82\)\( T^{13} + \)\(12\!\cdots\!71\)\( T^{14} - \)\(11\!\cdots\!36\)\( T^{15} + \)\(12\!\cdots\!71\)\( p^{3} T^{16} - \)\(82\!\cdots\!82\)\( p^{6} T^{17} + \)\(82\!\cdots\!27\)\( p^{9} T^{18} - 5631437161022373980 p^{12} T^{19} + 489560790263567776 p^{15} T^{20} - 3688624202139500 p^{18} T^{21} + 247104798829846 p^{21} T^{22} - 1881615778408 p^{24} T^{23} + 103542045132 p^{27} T^{24} - 890859104 p^{30} T^{25} + 37920506 p^{33} T^{26} - 353060 p^{36} T^{27} + 9693 p^{39} T^{28} - 62 p^{42} T^{29} + p^{45} T^{30} \) | |
| 13 | \( 1 - 148 T + 27952 T^{2} - 3059484 T^{3} + 358929801 T^{4} - 31878956230 T^{5} + 2888924063036 T^{6} - 218353050893794 T^{7} + 16524898299517484 T^{8} - 1091340288710616908 T^{9} + 71580893769923930318 T^{10} - \)\(41\!\cdots\!86\)\( T^{11} + \)\(24\!\cdots\!83\)\( T^{12} - \)\(12\!\cdots\!90\)\( T^{13} + \)\(66\!\cdots\!93\)\( T^{14} - \)\(31\!\cdots\!32\)\( T^{15} + \)\(66\!\cdots\!93\)\( p^{3} T^{16} - \)\(12\!\cdots\!90\)\( p^{6} T^{17} + \)\(24\!\cdots\!83\)\( p^{9} T^{18} - \)\(41\!\cdots\!86\)\( p^{12} T^{19} + 71580893769923930318 p^{15} T^{20} - 1091340288710616908 p^{18} T^{21} + 16524898299517484 p^{21} T^{22} - 218353050893794 p^{24} T^{23} + 2888924063036 p^{27} T^{24} - 31878956230 p^{30} T^{25} + 358929801 p^{33} T^{26} - 3059484 p^{36} T^{27} + 27952 p^{39} T^{28} - 148 p^{42} T^{29} + p^{45} T^{30} \) | |
| 17 | \( 1 - 132 T + 42773 T^{2} - 5315074 T^{3} + 939319843 T^{4} - 107413134002 T^{5} + 13942507151852 T^{6} - 1450487618140968 T^{7} + 155052666203058060 T^{8} - 14656690709502474898 T^{9} + \)\(13\!\cdots\!88\)\( T^{10} - \)\(11\!\cdots\!84\)\( T^{11} + \)\(97\!\cdots\!79\)\( T^{12} - \)\(76\!\cdots\!02\)\( T^{13} + \)\(57\!\cdots\!80\)\( T^{14} - \)\(41\!\cdots\!20\)\( T^{15} + \)\(57\!\cdots\!80\)\( p^{3} T^{16} - \)\(76\!\cdots\!02\)\( p^{6} T^{17} + \)\(97\!\cdots\!79\)\( p^{9} T^{18} - \)\(11\!\cdots\!84\)\( p^{12} T^{19} + \)\(13\!\cdots\!88\)\( p^{15} T^{20} - 14656690709502474898 p^{18} T^{21} + 155052666203058060 p^{21} T^{22} - 1450487618140968 p^{24} T^{23} + 13942507151852 p^{27} T^{24} - 107413134002 p^{30} T^{25} + 939319843 p^{33} T^{26} - 5315074 p^{36} T^{27} + 42773 p^{39} T^{28} - 132 p^{42} T^{29} + p^{45} T^{30} \) | |
| 19 | \( 1 - 260 T + 89531 T^{2} - 895294 p T^{3} + 3529324301 T^{4} - 544516357936 T^{5} + 86400394206199 T^{6} - 11393295442366108 T^{7} + 4169724888445898 p^{2} T^{8} - \)\(17\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!33\)\( T^{10} - \)\(20\!\cdots\!80\)\( T^{11} + \)\(20\!\cdots\!21\)\( T^{12} - \)\(53\!\cdots\!92\)\( p^{2} T^{13} + \)\(17\!\cdots\!82\)\( T^{14} - \)\(14\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!82\)\( p^{3} T^{16} - \)\(53\!\cdots\!92\)\( p^{8} T^{17} + \)\(20\!\cdots\!21\)\( p^{9} T^{18} - \)\(20\!\cdots\!80\)\( p^{12} T^{19} + \)\(19\!\cdots\!33\)\( p^{15} T^{20} - \)\(17\!\cdots\!24\)\( p^{18} T^{21} + 4169724888445898 p^{23} T^{22} - 11393295442366108 p^{24} T^{23} + 86400394206199 p^{27} T^{24} - 544516357936 p^{30} T^{25} + 3529324301 p^{33} T^{26} - 895294 p^{37} T^{27} + 89531 p^{39} T^{28} - 260 p^{42} T^{29} + p^{45} T^{30} \) | |
| 23 | \( 1 + 26 T + 111717 T^{2} + 2314968 T^{3} + 6185887719 T^{4} + 99853884478 T^{5} + 226645357709859 T^{6} + 2783429177200132 T^{7} + 6166614198969395136 T^{8} + 56617214123716224700 T^{9} + \)\(13\!\cdots\!79\)\( T^{10} + \)\(90\!\cdots\!30\)\( T^{11} + \)\(23\!\cdots\!83\)\( T^{12} + \)\(12\!\cdots\!68\)\( T^{13} + \)\(33\!\cdots\!88\)\( T^{14} + \)\(15\!\cdots\!76\)\( T^{15} + \)\(33\!\cdots\!88\)\( p^{3} T^{16} + \)\(12\!\cdots\!68\)\( p^{6} T^{17} + \)\(23\!\cdots\!83\)\( p^{9} T^{18} + \)\(90\!\cdots\!30\)\( p^{12} T^{19} + \)\(13\!\cdots\!79\)\( p^{15} T^{20} + 56617214123716224700 p^{18} T^{21} + 6166614198969395136 p^{21} T^{22} + 2783429177200132 p^{24} T^{23} + 226645357709859 p^{27} T^{24} + 99853884478 p^{30} T^{25} + 6185887719 p^{33} T^{26} + 2314968 p^{36} T^{27} + 111717 p^{39} T^{28} + 26 p^{42} T^{29} + p^{45} T^{30} \) | |
| 29 | \( 1 - 12 T + 212001 T^{2} + 706284 T^{3} + 21004079963 T^{4} + 311523943120 T^{5} + 1291561459148894 T^{6} + 28715889890363620 T^{7} + 55216710314608164598 T^{8} + \)\(14\!\cdots\!28\)\( T^{9} + \)\(17\!\cdots\!39\)\( T^{10} + \)\(43\!\cdots\!64\)\( T^{11} + \)\(44\!\cdots\!25\)\( T^{12} + \)\(96\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!43\)\( T^{14} + \)\(20\!\cdots\!64\)\( T^{15} + \)\(10\!\cdots\!43\)\( p^{3} T^{16} + \)\(96\!\cdots\!60\)\( p^{6} T^{17} + \)\(44\!\cdots\!25\)\( p^{9} T^{18} + \)\(43\!\cdots\!64\)\( p^{12} T^{19} + \)\(17\!\cdots\!39\)\( p^{15} T^{20} + \)\(14\!\cdots\!28\)\( p^{18} T^{21} + 55216710314608164598 p^{21} T^{22} + 28715889890363620 p^{24} T^{23} + 1291561459148894 p^{27} T^{24} + 311523943120 p^{30} T^{25} + 21004079963 p^{33} T^{26} + 706284 p^{36} T^{27} + 212001 p^{39} T^{28} - 12 p^{42} T^{29} + p^{45} T^{30} \) | |
| 31 | \( 1 - 144 T + 173367 T^{2} - 30017190 T^{3} + 15825949141 T^{4} - 2795557847714 T^{5} + 1014623744997456 T^{6} - 170236995138375994 T^{7} + 50245667053835168594 T^{8} - \)\(80\!\cdots\!22\)\( T^{9} + \)\(20\!\cdots\!29\)\( T^{10} - \)\(32\!\cdots\!30\)\( T^{11} + \)\(73\!\cdots\!67\)\( T^{12} - \)\(11\!\cdots\!12\)\( T^{13} + \)\(24\!\cdots\!39\)\( T^{14} - \)\(35\!\cdots\!84\)\( T^{15} + \)\(24\!\cdots\!39\)\( p^{3} T^{16} - \)\(11\!\cdots\!12\)\( p^{6} T^{17} + \)\(73\!\cdots\!67\)\( p^{9} T^{18} - \)\(32\!\cdots\!30\)\( p^{12} T^{19} + \)\(20\!\cdots\!29\)\( p^{15} T^{20} - \)\(80\!\cdots\!22\)\( p^{18} T^{21} + 50245667053835168594 p^{21} T^{22} - 170236995138375994 p^{24} T^{23} + 1014623744997456 p^{27} T^{24} - 2795557847714 p^{30} T^{25} + 15825949141 p^{33} T^{26} - 30017190 p^{36} T^{27} + 173367 p^{39} T^{28} - 144 p^{42} T^{29} + p^{45} T^{30} \) | |
| 37 | \( 1 - 274 T + 398492 T^{2} - 87356782 T^{3} + 79816118904 T^{4} - 14635435475978 T^{5} + 10756449537193740 T^{6} - 1678361860276017328 T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(14\!\cdots\!98\)\( T^{9} + \)\(89\!\cdots\!68\)\( T^{10} - \)\(10\!\cdots\!02\)\( T^{11} + \)\(61\!\cdots\!92\)\( T^{12} - \)\(64\!\cdots\!34\)\( T^{13} + \)\(36\!\cdots\!81\)\( T^{14} - \)\(34\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!81\)\( p^{3} T^{16} - \)\(64\!\cdots\!34\)\( p^{6} T^{17} + \)\(61\!\cdots\!92\)\( p^{9} T^{18} - \)\(10\!\cdots\!02\)\( p^{12} T^{19} + \)\(89\!\cdots\!68\)\( p^{15} T^{20} - \)\(14\!\cdots\!98\)\( p^{18} T^{21} + \)\(10\!\cdots\!06\)\( p^{21} T^{22} - 1678361860276017328 p^{24} T^{23} + 10756449537193740 p^{27} T^{24} - 14635435475978 p^{30} T^{25} + 79816118904 p^{33} T^{26} - 87356782 p^{36} T^{27} + 398492 p^{39} T^{28} - 274 p^{42} T^{29} + p^{45} T^{30} \) | |
| 43 | \( 1 - 986 T + 991223 T^{2} - 672640622 T^{3} + 431413126321 T^{4} - 231952472283980 T^{5} + 117113071623247482 T^{6} - 53188472501703805680 T^{7} + \)\(22\!\cdots\!34\)\( T^{8} - \)\(90\!\cdots\!84\)\( T^{9} + \)\(33\!\cdots\!42\)\( T^{10} - \)\(12\!\cdots\!60\)\( T^{11} + \)\(40\!\cdots\!93\)\( T^{12} - \)\(12\!\cdots\!58\)\( T^{13} + \)\(39\!\cdots\!46\)\( T^{14} - \)\(11\!\cdots\!88\)\( T^{15} + \)\(39\!\cdots\!46\)\( p^{3} T^{16} - \)\(12\!\cdots\!58\)\( p^{6} T^{17} + \)\(40\!\cdots\!93\)\( p^{9} T^{18} - \)\(12\!\cdots\!60\)\( p^{12} T^{19} + \)\(33\!\cdots\!42\)\( p^{15} T^{20} - \)\(90\!\cdots\!84\)\( p^{18} T^{21} + \)\(22\!\cdots\!34\)\( p^{21} T^{22} - 53188472501703805680 p^{24} T^{23} + 117113071623247482 p^{27} T^{24} - 231952472283980 p^{30} T^{25} + 431413126321 p^{33} T^{26} - 672640622 p^{36} T^{27} + 991223 p^{39} T^{28} - 986 p^{42} T^{29} + p^{45} T^{30} \) | |
| 47 | \( 1 - 44 T + 791138 T^{2} - 22944728 T^{3} + 325875865252 T^{4} - 8984805584614 T^{5} + 91839304950645092 T^{6} - 2894783306665103392 T^{7} + \)\(19\!\cdots\!46\)\( T^{8} - \)\(72\!\cdots\!98\)\( T^{9} + \)\(33\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!52\)\( T^{11} + \)\(48\!\cdots\!42\)\( T^{12} - \)\(20\!\cdots\!84\)\( T^{13} + \)\(58\!\cdots\!83\)\( T^{14} - \)\(24\!\cdots\!64\)\( T^{15} + \)\(58\!\cdots\!83\)\( p^{3} T^{16} - \)\(20\!\cdots\!84\)\( p^{6} T^{17} + \)\(48\!\cdots\!42\)\( p^{9} T^{18} - \)\(14\!\cdots\!52\)\( p^{12} T^{19} + \)\(33\!\cdots\!64\)\( p^{15} T^{20} - \)\(72\!\cdots\!98\)\( p^{18} T^{21} + \)\(19\!\cdots\!46\)\( p^{21} T^{22} - 2894783306665103392 p^{24} T^{23} + 91839304950645092 p^{27} T^{24} - 8984805584614 p^{30} T^{25} + 325875865252 p^{33} T^{26} - 22944728 p^{36} T^{27} + 791138 p^{39} T^{28} - 44 p^{42} T^{29} + p^{45} T^{30} \) | |
| 53 | \( 1 + 366 T + 1425137 T^{2} + 491983896 T^{3} + 1038844943267 T^{4} + 335810959159586 T^{5} + 504031609036134634 T^{6} + \)\(15\!\cdots\!82\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} + \)\(50\!\cdots\!46\)\( T^{9} + \)\(50\!\cdots\!07\)\( T^{10} + \)\(13\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!37\)\( T^{12} + \)\(26\!\cdots\!22\)\( T^{13} + \)\(20\!\cdots\!59\)\( T^{14} + \)\(44\!\cdots\!80\)\( T^{15} + \)\(20\!\cdots\!59\)\( p^{3} T^{16} + \)\(26\!\cdots\!22\)\( p^{6} T^{17} + \)\(11\!\cdots\!37\)\( p^{9} T^{18} + \)\(13\!\cdots\!20\)\( p^{12} T^{19} + \)\(50\!\cdots\!07\)\( p^{15} T^{20} + \)\(50\!\cdots\!46\)\( p^{18} T^{21} + \)\(18\!\cdots\!26\)\( p^{21} T^{22} + \)\(15\!\cdots\!82\)\( p^{24} T^{23} + 504031609036134634 p^{27} T^{24} + 335810959159586 p^{30} T^{25} + 1038844943267 p^{33} T^{26} + 491983896 p^{36} T^{27} + 1425137 p^{39} T^{28} + 366 p^{42} T^{29} + p^{45} T^{30} \) | |
| 59 | \( 1 + 8 T + 2180472 T^{2} + 67689980 T^{3} + 2314550260985 T^{4} + 119520353931424 T^{5} + 1589042705204621296 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(79\!\cdots\!98\)\( T^{8} + \)\(64\!\cdots\!76\)\( T^{9} + \)\(51\!\cdots\!77\)\( p T^{10} + \)\(27\!\cdots\!88\)\( T^{11} + \)\(92\!\cdots\!46\)\( T^{12} + \)\(83\!\cdots\!20\)\( T^{13} + \)\(23\!\cdots\!19\)\( T^{14} + \)\(33\!\cdots\!60\)\( p T^{15} + \)\(23\!\cdots\!19\)\( p^{3} T^{16} + \)\(83\!\cdots\!20\)\( p^{6} T^{17} + \)\(92\!\cdots\!46\)\( p^{9} T^{18} + \)\(27\!\cdots\!88\)\( p^{12} T^{19} + \)\(51\!\cdots\!77\)\( p^{16} T^{20} + \)\(64\!\cdots\!76\)\( p^{18} T^{21} + \)\(79\!\cdots\!98\)\( p^{21} T^{22} + \)\(10\!\cdots\!12\)\( p^{24} T^{23} + 1589042705204621296 p^{27} T^{24} + 119520353931424 p^{30} T^{25} + 2314550260985 p^{33} T^{26} + 67689980 p^{36} T^{27} + 2180472 p^{39} T^{28} + 8 p^{42} T^{29} + p^{45} T^{30} \) | |
| 61 | \( 1 - 1396 T + 3020080 T^{2} - 3282569402 T^{3} + 4058456672171 T^{4} - 3598286361228136 T^{5} + 3284892311308547671 T^{6} - \)\(24\!\cdots\!90\)\( T^{7} + \)\(18\!\cdots\!63\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(74\!\cdots\!97\)\( T^{10} - \)\(42\!\cdots\!06\)\( T^{11} + \)\(24\!\cdots\!36\)\( T^{12} - \)\(12\!\cdots\!04\)\( T^{13} + \)\(64\!\cdots\!57\)\( T^{14} - \)\(50\!\cdots\!44\)\( p T^{15} + \)\(64\!\cdots\!57\)\( p^{3} T^{16} - \)\(12\!\cdots\!04\)\( p^{6} T^{17} + \)\(24\!\cdots\!36\)\( p^{9} T^{18} - \)\(42\!\cdots\!06\)\( p^{12} T^{19} + \)\(74\!\cdots\!97\)\( p^{15} T^{20} - \)\(11\!\cdots\!52\)\( p^{18} T^{21} + \)\(18\!\cdots\!63\)\( p^{21} T^{22} - \)\(24\!\cdots\!90\)\( p^{24} T^{23} + 3284892311308547671 p^{27} T^{24} - 3598286361228136 p^{30} T^{25} + 4058456672171 p^{33} T^{26} - 3282569402 p^{36} T^{27} + 3020080 p^{39} T^{28} - 1396 p^{42} T^{29} + p^{45} T^{30} \) | |
| 67 | \( 1 + 24 T + 2756606 T^{2} - 43610100 T^{3} + 3783735888671 T^{4} - 195698680587882 T^{5} + 3429474215860355820 T^{6} - \)\(28\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!98\)\( T^{9} + \)\(12\!\cdots\!89\)\( T^{10} - \)\(14\!\cdots\!48\)\( T^{11} + \)\(52\!\cdots\!48\)\( T^{12} - \)\(64\!\cdots\!80\)\( T^{13} + \)\(18\!\cdots\!19\)\( T^{14} - \)\(22\!\cdots\!64\)\( T^{15} + \)\(18\!\cdots\!19\)\( p^{3} T^{16} - \)\(64\!\cdots\!80\)\( p^{6} T^{17} + \)\(52\!\cdots\!48\)\( p^{9} T^{18} - \)\(14\!\cdots\!48\)\( p^{12} T^{19} + \)\(12\!\cdots\!89\)\( p^{15} T^{20} - \)\(24\!\cdots\!98\)\( p^{18} T^{21} + \)\(23\!\cdots\!06\)\( p^{21} T^{22} - \)\(28\!\cdots\!76\)\( p^{24} T^{23} + 3429474215860355820 p^{27} T^{24} - 195698680587882 p^{30} T^{25} + 3783735888671 p^{33} T^{26} - 43610100 p^{36} T^{27} + 2756606 p^{39} T^{28} + 24 p^{42} T^{29} + p^{45} T^{30} \) | |
| 71 | \( 1 - 1464 T + 4754606 T^{2} - 5355170276 T^{3} + 9952409477611 T^{4} - 9180748202405608 T^{5} + 12652891636898997797 T^{6} - \)\(99\!\cdots\!24\)\( T^{7} + \)\(11\!\cdots\!25\)\( T^{8} - \)\(76\!\cdots\!92\)\( T^{9} + \)\(75\!\cdots\!61\)\( T^{10} - \)\(45\!\cdots\!20\)\( T^{11} + \)\(39\!\cdots\!68\)\( T^{12} - \)\(21\!\cdots\!12\)\( T^{13} + \)\(17\!\cdots\!53\)\( T^{14} - \)\(85\!\cdots\!00\)\( T^{15} + \)\(17\!\cdots\!53\)\( p^{3} T^{16} - \)\(21\!\cdots\!12\)\( p^{6} T^{17} + \)\(39\!\cdots\!68\)\( p^{9} T^{18} - \)\(45\!\cdots\!20\)\( p^{12} T^{19} + \)\(75\!\cdots\!61\)\( p^{15} T^{20} - \)\(76\!\cdots\!92\)\( p^{18} T^{21} + \)\(11\!\cdots\!25\)\( p^{21} T^{22} - \)\(99\!\cdots\!24\)\( p^{24} T^{23} + 12652891636898997797 p^{27} T^{24} - 9180748202405608 p^{30} T^{25} + 9952409477611 p^{33} T^{26} - 5355170276 p^{36} T^{27} + 4754606 p^{39} T^{28} - 1464 p^{42} T^{29} + p^{45} T^{30} \) | |
| 73 | \( 1 - 1174 T + 4693261 T^{2} - 5016159180 T^{3} + 10640032200024 T^{4} - 10308277693737844 T^{5} + 15431447793244833782 T^{6} - \)\(13\!\cdots\!04\)\( T^{7} + \)\(15\!\cdots\!84\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!10\)\( T^{10} - \)\(89\!\cdots\!28\)\( T^{11} + \)\(76\!\cdots\!09\)\( T^{12} - \)\(49\!\cdots\!42\)\( T^{13} + \)\(37\!\cdots\!29\)\( T^{14} - \)\(21\!\cdots\!72\)\( T^{15} + \)\(37\!\cdots\!29\)\( p^{3} T^{16} - \)\(49\!\cdots\!42\)\( p^{6} T^{17} + \)\(76\!\cdots\!09\)\( p^{9} T^{18} - \)\(89\!\cdots\!28\)\( p^{12} T^{19} + \)\(12\!\cdots\!10\)\( p^{15} T^{20} - \)\(12\!\cdots\!52\)\( p^{18} T^{21} + \)\(15\!\cdots\!84\)\( p^{21} T^{22} - \)\(13\!\cdots\!04\)\( p^{24} T^{23} + 15431447793244833782 p^{27} T^{24} - 10308277693737844 p^{30} T^{25} + 10640032200024 p^{33} T^{26} - 5016159180 p^{36} T^{27} + 4693261 p^{39} T^{28} - 1174 p^{42} T^{29} + p^{45} T^{30} \) | |
| 79 | \( 1 - 1590 T + 3782676 T^{2} - 4860458350 T^{3} + 6652851448462 T^{4} - 6934357283178538 T^{5} + 7142749414969129439 T^{6} - \)\(61\!\cdots\!76\)\( T^{7} + \)\(53\!\cdots\!63\)\( T^{8} - \)\(39\!\cdots\!26\)\( T^{9} + \)\(30\!\cdots\!30\)\( T^{10} - \)\(21\!\cdots\!86\)\( T^{11} + \)\(16\!\cdots\!12\)\( T^{12} - \)\(10\!\cdots\!90\)\( T^{13} + \)\(81\!\cdots\!33\)\( T^{14} - \)\(54\!\cdots\!96\)\( T^{15} + \)\(81\!\cdots\!33\)\( p^{3} T^{16} - \)\(10\!\cdots\!90\)\( p^{6} T^{17} + \)\(16\!\cdots\!12\)\( p^{9} T^{18} - \)\(21\!\cdots\!86\)\( p^{12} T^{19} + \)\(30\!\cdots\!30\)\( p^{15} T^{20} - \)\(39\!\cdots\!26\)\( p^{18} T^{21} + \)\(53\!\cdots\!63\)\( p^{21} T^{22} - \)\(61\!\cdots\!76\)\( p^{24} T^{23} + 7142749414969129439 p^{27} T^{24} - 6934357283178538 p^{30} T^{25} + 6652851448462 p^{33} T^{26} - 4860458350 p^{36} T^{27} + 3782676 p^{39} T^{28} - 1590 p^{42} T^{29} + p^{45} T^{30} \) | |
| 83 | \( 1 - 1970 T + 6024661 T^{2} - 9762224988 T^{3} + 18068173432790 T^{4} - 24694100911608080 T^{5} + 35277324428288689492 T^{6} - \)\(41\!\cdots\!00\)\( T^{7} + \)\(50\!\cdots\!90\)\( T^{8} - \)\(52\!\cdots\!80\)\( T^{9} + \)\(54\!\cdots\!68\)\( T^{10} - \)\(51\!\cdots\!52\)\( T^{11} + \)\(47\!\cdots\!95\)\( T^{12} - \)\(39\!\cdots\!46\)\( T^{13} + \)\(33\!\cdots\!27\)\( T^{14} - \)\(25\!\cdots\!40\)\( T^{15} + \)\(33\!\cdots\!27\)\( p^{3} T^{16} - \)\(39\!\cdots\!46\)\( p^{6} T^{17} + \)\(47\!\cdots\!95\)\( p^{9} T^{18} - \)\(51\!\cdots\!52\)\( p^{12} T^{19} + \)\(54\!\cdots\!68\)\( p^{15} T^{20} - \)\(52\!\cdots\!80\)\( p^{18} T^{21} + \)\(50\!\cdots\!90\)\( p^{21} T^{22} - \)\(41\!\cdots\!00\)\( p^{24} T^{23} + 35277324428288689492 p^{27} T^{24} - 24694100911608080 p^{30} T^{25} + 18068173432790 p^{33} T^{26} - 9762224988 p^{36} T^{27} + 6024661 p^{39} T^{28} - 1970 p^{42} T^{29} + p^{45} T^{30} \) | |
| 89 | \( 1 - 1972 T + 5305590 T^{2} - 8891974112 T^{3} + 14219265945708 T^{4} - 19962059887580286 T^{5} + 25658856868831696551 T^{6} - \)\(30\!\cdots\!06\)\( T^{7} + \)\(35\!\cdots\!06\)\( T^{8} - \)\(37\!\cdots\!62\)\( T^{9} + \)\(39\!\cdots\!47\)\( T^{10} - \)\(38\!\cdots\!70\)\( T^{11} + \)\(37\!\cdots\!52\)\( T^{12} - \)\(38\!\cdots\!62\)\( p T^{13} + \)\(30\!\cdots\!89\)\( T^{14} - \)\(25\!\cdots\!36\)\( T^{15} + \)\(30\!\cdots\!89\)\( p^{3} T^{16} - \)\(38\!\cdots\!62\)\( p^{7} T^{17} + \)\(37\!\cdots\!52\)\( p^{9} T^{18} - \)\(38\!\cdots\!70\)\( p^{12} T^{19} + \)\(39\!\cdots\!47\)\( p^{15} T^{20} - \)\(37\!\cdots\!62\)\( p^{18} T^{21} + \)\(35\!\cdots\!06\)\( p^{21} T^{22} - \)\(30\!\cdots\!06\)\( p^{24} T^{23} + 25658856868831696551 p^{27} T^{24} - 19962059887580286 p^{30} T^{25} + 14219265945708 p^{33} T^{26} - 8891974112 p^{36} T^{27} + 5305590 p^{39} T^{28} - 1972 p^{42} T^{29} + p^{45} T^{30} \) | |
| 97 | \( 1 - 954 T + 8736264 T^{2} - 8656647752 T^{3} + 37724131119776 T^{4} - 37707983714354192 T^{5} + 1109072269011285525 p T^{6} - \)\(10\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!30\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{9} + \)\(37\!\cdots\!39\)\( T^{10} - \)\(33\!\cdots\!54\)\( T^{11} + \)\(50\!\cdots\!54\)\( T^{12} - \)\(41\!\cdots\!68\)\( T^{13} + \)\(55\!\cdots\!75\)\( T^{14} - \)\(41\!\cdots\!36\)\( T^{15} + \)\(55\!\cdots\!75\)\( p^{3} T^{16} - \)\(41\!\cdots\!68\)\( p^{6} T^{17} + \)\(50\!\cdots\!54\)\( p^{9} T^{18} - \)\(33\!\cdots\!54\)\( p^{12} T^{19} + \)\(37\!\cdots\!39\)\( p^{15} T^{20} - \)\(21\!\cdots\!00\)\( p^{18} T^{21} + \)\(22\!\cdots\!30\)\( p^{21} T^{22} - \)\(10\!\cdots\!64\)\( p^{24} T^{23} + 1109072269011285525 p^{28} T^{24} - 37707983714354192 p^{30} T^{25} + 37724131119776 p^{33} T^{26} - 8656647752 p^{36} T^{27} + 8736264 p^{39} T^{28} - 954 p^{42} T^{29} + p^{45} T^{30} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.09531822234914247548447613250, −2.07338540108645871700712511286, −2.07257229158117344909559627110, −2.04364570537749550300447077929, −2.01176683893384354384539843251, −2.00475611205970433501762219989, −1.90699476852567147780451459443, −1.68255554108829049087920222233, −1.58305400261854358529726797853, −1.55304567271249631720748617567, −1.43763191306025049851992851142, −1.23682253883057946217151359617, −1.11842747692810932539920455773, −1.10020655579964389319106351595, −1.02971724641968116101659444039, −0.889840889013309515061229985994, −0.856672548705675130966810180439, −0.816494339777616145060669008941, −0.71285673238689625850332656577, −0.65897116866592873825037942428, −0.59714061712318976151473667309, −0.45562518305842085141851438129, −0.44033420682202244127923772250, −0.43953993823576415079508070258, −0.20599212055619424103938416470, 0.20599212055619424103938416470, 0.43953993823576415079508070258, 0.44033420682202244127923772250, 0.45562518305842085141851438129, 0.59714061712318976151473667309, 0.65897116866592873825037942428, 0.71285673238689625850332656577, 0.816494339777616145060669008941, 0.856672548705675130966810180439, 0.889840889013309515061229985994, 1.02971724641968116101659444039, 1.10020655579964389319106351595, 1.11842747692810932539920455773, 1.23682253883057946217151359617, 1.43763191306025049851992851142, 1.55304567271249631720748617567, 1.58305400261854358529726797853, 1.68255554108829049087920222233, 1.90699476852567147780451459443, 2.00475611205970433501762219989, 2.01176683893384354384539843251, 2.04364570537749550300447077929, 2.07257229158117344909559627110, 2.07338540108645871700712511286, 2.09531822234914247548447613250
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.