Dirichlet series
| L(s) = 1 | − 6·3-s − 43·4-s + 24·5-s + 62·7-s + 7·8-s − 144·9-s − 187·11-s + 258·12-s − 144·15-s + 856·16-s − 74·17-s + 159·19-s − 1.03e3·20-s − 372·21-s − 215·23-s − 42·24-s − 727·25-s + 926·27-s − 2.66e3·28-s − 157·29-s + 394·31-s − 224·32-s + 1.12e3·33-s + 1.48e3·35-s + 6.19e3·36-s − 88·37-s + 168·40-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 5.37·4-s + 2.14·5-s + 3.34·7-s + 0.309·8-s − 5.33·9-s − 5.12·11-s + 6.20·12-s − 2.47·15-s + 13.3·16-s − 1.05·17-s + 1.91·19-s − 11.5·20-s − 3.86·21-s − 1.94·23-s − 0.357·24-s − 5.81·25-s + 6.60·27-s − 17.9·28-s − 1.00·29-s + 2.28·31-s − 1.23·32-s + 5.91·33-s + 7.18·35-s + 86/3·36-s − 0.391·37-s + 0.664·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{17} \cdot 13^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{17} \cdot 13^{34}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(11^{17} \cdot 13^{34}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.81363\times 10^{34}\) |
| Root analytic conductor: | \(10.4730\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((34,\ 11^{17} \cdot 13^{34} ,\ ( \ : [3/2]^{17} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.006082509272\) |
| \(L(\frac12)\) | \(\approx\) | \(0.006082509272\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + p T )^{17} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 43 T^{2} - 7 T^{3} + 993 T^{4} - 189 p T^{5} + 8327 p T^{6} - 4867 p T^{7} + 228475 T^{8} - 41365 p^{2} T^{9} + 2710231 T^{10} - 2110771 T^{11} + 28689595 T^{12} - 11045309 p T^{13} + 34416187 p^{3} T^{14} - 25320815 p^{3} T^{15} + 9413099 p^{8} T^{16} - 3302273 p^{9} T^{17} + 9413099 p^{11} T^{18} - 25320815 p^{9} T^{19} + 34416187 p^{12} T^{20} - 11045309 p^{13} T^{21} + 28689595 p^{15} T^{22} - 2110771 p^{18} T^{23} + 2710231 p^{21} T^{24} - 41365 p^{26} T^{25} + 228475 p^{27} T^{26} - 4867 p^{31} T^{27} + 8327 p^{34} T^{28} - 189 p^{37} T^{29} + 993 p^{39} T^{30} - 7 p^{42} T^{31} + 43 p^{45} T^{32} + p^{51} T^{34} \) |
| 3 | \( 1 + 2 p T + 20 p^{2} T^{2} + 1018 T^{3} + 209 p^{4} T^{4} + 93722 T^{5} + 1118030 T^{6} + 6085216 T^{7} + 57972610 T^{8} + 308364392 T^{9} + 2507579080 T^{10} + 4310876624 p T^{11} + 31277552848 p T^{12} + 465870532574 T^{13} + 3108670756847 T^{14} + 4931331206114 p T^{15} + 92612903143025 T^{16} + 420900884370772 T^{17} + 92612903143025 p^{3} T^{18} + 4931331206114 p^{7} T^{19} + 3108670756847 p^{9} T^{20} + 465870532574 p^{12} T^{21} + 31277552848 p^{16} T^{22} + 4310876624 p^{19} T^{23} + 2507579080 p^{21} T^{24} + 308364392 p^{24} T^{25} + 57972610 p^{27} T^{26} + 6085216 p^{30} T^{27} + 1118030 p^{33} T^{28} + 93722 p^{36} T^{29} + 209 p^{43} T^{30} + 1018 p^{42} T^{31} + 20 p^{47} T^{32} + 2 p^{49} T^{33} + p^{51} T^{34} \) | |
| 5 | \( 1 - 24 T + 1303 T^{2} - 25646 T^{3} + 161719 p T^{4} - 13735682 T^{5} + 65305274 p T^{6} - 4936264816 T^{7} + 19514034841 p T^{8} - 1337799957256 T^{9} + 23100361837153 T^{10} - 290928835302318 T^{11} + 4518652694635297 T^{12} - 52737691223675712 T^{13} + 750544279832525739 T^{14} - 1632761827880575558 p T^{15} + 21538361079972089714 p T^{16} - 43768201899611863724 p^{2} T^{17} + 21538361079972089714 p^{4} T^{18} - 1632761827880575558 p^{7} T^{19} + 750544279832525739 p^{9} T^{20} - 52737691223675712 p^{12} T^{21} + 4518652694635297 p^{15} T^{22} - 290928835302318 p^{18} T^{23} + 23100361837153 p^{21} T^{24} - 1337799957256 p^{24} T^{25} + 19514034841 p^{28} T^{26} - 4936264816 p^{30} T^{27} + 65305274 p^{34} T^{28} - 13735682 p^{36} T^{29} + 161719 p^{40} T^{30} - 25646 p^{42} T^{31} + 1303 p^{45} T^{32} - 24 p^{48} T^{33} + p^{51} T^{34} \) | |
| 7 | \( 1 - 62 T + 80 p^{2} T^{2} - 160060 T^{3} + 6076563 T^{4} - 186287018 T^{5} + 5265193702 T^{6} - 128286519670 T^{7} + 2892342475500 T^{8} - 57171723324084 T^{9} + 1046210242192854 T^{10} - 2348086199806806 p T^{11} + 233100245695006604 T^{12} - 347841223170177810 p T^{13} + 15708341081906265977 T^{14} + \)\(25\!\cdots\!24\)\( T^{15} - \)\(98\!\cdots\!83\)\( T^{16} + \)\(23\!\cdots\!08\)\( T^{17} - \)\(98\!\cdots\!83\)\( p^{3} T^{18} + \)\(25\!\cdots\!24\)\( p^{6} T^{19} + 15708341081906265977 p^{9} T^{20} - 347841223170177810 p^{13} T^{21} + 233100245695006604 p^{15} T^{22} - 2348086199806806 p^{19} T^{23} + 1046210242192854 p^{21} T^{24} - 57171723324084 p^{24} T^{25} + 2892342475500 p^{27} T^{26} - 128286519670 p^{30} T^{27} + 5265193702 p^{33} T^{28} - 186287018 p^{36} T^{29} + 6076563 p^{39} T^{30} - 160060 p^{42} T^{31} + 80 p^{47} T^{32} - 62 p^{48} T^{33} + p^{51} T^{34} \) | |
| 17 | \( 1 + 74 T + 42525 T^{2} + 2399590 T^{3} + 849871071 T^{4} + 38717555592 T^{5} + 11276049856428 T^{6} + 452486903923644 T^{7} + 115787258551306431 T^{8} + 4399158107159922918 T^{9} + \)\(98\!\cdots\!29\)\( T^{10} + \)\(36\!\cdots\!06\)\( T^{11} + \)\(70\!\cdots\!15\)\( T^{12} + \)\(25\!\cdots\!68\)\( T^{13} + \)\(44\!\cdots\!85\)\( T^{14} + \)\(15\!\cdots\!34\)\( T^{15} + \)\(24\!\cdots\!42\)\( T^{16} + \)\(81\!\cdots\!48\)\( T^{17} + \)\(24\!\cdots\!42\)\( p^{3} T^{18} + \)\(15\!\cdots\!34\)\( p^{6} T^{19} + \)\(44\!\cdots\!85\)\( p^{9} T^{20} + \)\(25\!\cdots\!68\)\( p^{12} T^{21} + \)\(70\!\cdots\!15\)\( p^{15} T^{22} + \)\(36\!\cdots\!06\)\( p^{18} T^{23} + \)\(98\!\cdots\!29\)\( p^{21} T^{24} + 4399158107159922918 p^{24} T^{25} + 115787258551306431 p^{27} T^{26} + 452486903923644 p^{30} T^{27} + 11276049856428 p^{33} T^{28} + 38717555592 p^{36} T^{29} + 849871071 p^{39} T^{30} + 2399590 p^{42} T^{31} + 42525 p^{45} T^{32} + 74 p^{48} T^{33} + p^{51} T^{34} \) | |
| 19 | \( 1 - 159 T + 64043 T^{2} - 9913904 T^{3} + 113072999 p T^{4} - 303572857355 T^{5} + 48455871457646 T^{6} - 6148679457340507 T^{7} + 808743656935085565 T^{8} - 92763454104719713572 T^{9} + \)\(10\!\cdots\!03\)\( T^{10} - \)\(11\!\cdots\!51\)\( T^{11} + \)\(59\!\cdots\!96\)\( p T^{12} - \)\(10\!\cdots\!89\)\( T^{13} + \)\(10\!\cdots\!63\)\( T^{14} - \)\(92\!\cdots\!50\)\( T^{15} + \)\(80\!\cdots\!04\)\( T^{16} - \)\(68\!\cdots\!42\)\( T^{17} + \)\(80\!\cdots\!04\)\( p^{3} T^{18} - \)\(92\!\cdots\!50\)\( p^{6} T^{19} + \)\(10\!\cdots\!63\)\( p^{9} T^{20} - \)\(10\!\cdots\!89\)\( p^{12} T^{21} + \)\(59\!\cdots\!96\)\( p^{16} T^{22} - \)\(11\!\cdots\!51\)\( p^{18} T^{23} + \)\(10\!\cdots\!03\)\( p^{21} T^{24} - 92763454104719713572 p^{24} T^{25} + 808743656935085565 p^{27} T^{26} - 6148679457340507 p^{30} T^{27} + 48455871457646 p^{33} T^{28} - 303572857355 p^{36} T^{29} + 113072999 p^{40} T^{30} - 9913904 p^{42} T^{31} + 64043 p^{45} T^{32} - 159 p^{48} T^{33} + p^{51} T^{34} \) | |
| 23 | \( 1 + 215 T + 114284 T^{2} + 16576291 T^{3} + 5546705417 T^{4} + 592876924712 T^{5} + 171782862467068 T^{6} + 14288609511675680 T^{7} + 4119211220887728412 T^{8} + \)\(27\!\cdots\!66\)\( T^{9} + \)\(83\!\cdots\!26\)\( T^{10} + \)\(46\!\cdots\!18\)\( T^{11} + \)\(14\!\cdots\!58\)\( T^{12} + \)\(68\!\cdots\!68\)\( T^{13} + \)\(22\!\cdots\!95\)\( T^{14} + \)\(93\!\cdots\!35\)\( T^{15} + \)\(30\!\cdots\!53\)\( T^{16} + \)\(11\!\cdots\!78\)\( T^{17} + \)\(30\!\cdots\!53\)\( p^{3} T^{18} + \)\(93\!\cdots\!35\)\( p^{6} T^{19} + \)\(22\!\cdots\!95\)\( p^{9} T^{20} + \)\(68\!\cdots\!68\)\( p^{12} T^{21} + \)\(14\!\cdots\!58\)\( p^{15} T^{22} + \)\(46\!\cdots\!18\)\( p^{18} T^{23} + \)\(83\!\cdots\!26\)\( p^{21} T^{24} + \)\(27\!\cdots\!66\)\( p^{24} T^{25} + 4119211220887728412 p^{27} T^{26} + 14288609511675680 p^{30} T^{27} + 171782862467068 p^{33} T^{28} + 592876924712 p^{36} T^{29} + 5546705417 p^{39} T^{30} + 16576291 p^{42} T^{31} + 114284 p^{45} T^{32} + 215 p^{48} T^{33} + p^{51} T^{34} \) | |
| 29 | \( 1 + 157 T + 218122 T^{2} + 30119405 T^{3} + 23733605348 T^{4} + 3015688758063 T^{5} + 1738016018799934 T^{6} + 209096911548074483 T^{7} + 3336193540741237133 p T^{8} + \)\(11\!\cdots\!58\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} + \)\(48\!\cdots\!74\)\( T^{11} + \)\(16\!\cdots\!38\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{13} + \)\(53\!\cdots\!60\)\( T^{14} + \)\(53\!\cdots\!92\)\( T^{15} + \)\(14\!\cdots\!33\)\( T^{16} + \)\(14\!\cdots\!45\)\( T^{17} + \)\(14\!\cdots\!33\)\( p^{3} T^{18} + \)\(53\!\cdots\!92\)\( p^{6} T^{19} + \)\(53\!\cdots\!60\)\( p^{9} T^{20} + \)\(17\!\cdots\!04\)\( p^{12} T^{21} + \)\(16\!\cdots\!38\)\( p^{15} T^{22} + \)\(48\!\cdots\!74\)\( p^{18} T^{23} + \)\(43\!\cdots\!72\)\( p^{21} T^{24} + \)\(11\!\cdots\!58\)\( p^{24} T^{25} + 3336193540741237133 p^{28} T^{26} + 209096911548074483 p^{30} T^{27} + 1738016018799934 p^{33} T^{28} + 3015688758063 p^{36} T^{29} + 23733605348 p^{39} T^{30} + 30119405 p^{42} T^{31} + 218122 p^{45} T^{32} + 157 p^{48} T^{33} + p^{51} T^{34} \) | |
| 31 | \( 1 - 394 T + 317766 T^{2} - 95219372 T^{3} + 44620166446 T^{4} - 10737961760138 T^{5} + 3783765662426933 T^{6} - 24094068512466860 p T^{7} + \)\(21\!\cdots\!94\)\( T^{8} - \)\(35\!\cdots\!32\)\( T^{9} + \)\(92\!\cdots\!56\)\( T^{10} - \)\(11\!\cdots\!36\)\( T^{11} + \)\(29\!\cdots\!47\)\( T^{12} - \)\(25\!\cdots\!18\)\( T^{13} + \)\(75\!\cdots\!00\)\( T^{14} - \)\(33\!\cdots\!24\)\( T^{15} + \)\(18\!\cdots\!71\)\( T^{16} - \)\(39\!\cdots\!96\)\( T^{17} + \)\(18\!\cdots\!71\)\( p^{3} T^{18} - \)\(33\!\cdots\!24\)\( p^{6} T^{19} + \)\(75\!\cdots\!00\)\( p^{9} T^{20} - \)\(25\!\cdots\!18\)\( p^{12} T^{21} + \)\(29\!\cdots\!47\)\( p^{15} T^{22} - \)\(11\!\cdots\!36\)\( p^{18} T^{23} + \)\(92\!\cdots\!56\)\( p^{21} T^{24} - \)\(35\!\cdots\!32\)\( p^{24} T^{25} + \)\(21\!\cdots\!94\)\( p^{27} T^{26} - 24094068512466860 p^{31} T^{27} + 3783765662426933 p^{33} T^{28} - 10737961760138 p^{36} T^{29} + 44620166446 p^{39} T^{30} - 95219372 p^{42} T^{31} + 317766 p^{45} T^{32} - 394 p^{48} T^{33} + p^{51} T^{34} \) | |
| 37 | \( 1 + 88 T + 484512 T^{2} + 37370988 T^{3} + 115681778818 T^{4} + 7212382055440 T^{5} + 18048779405752596 T^{6} + 799167654299434508 T^{7} + \)\(20\!\cdots\!43\)\( T^{8} + \)\(48\!\cdots\!90\)\( T^{9} + \)\(18\!\cdots\!32\)\( T^{10} + \)\(25\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!78\)\( T^{12} - \)\(26\!\cdots\!96\)\( T^{13} + \)\(85\!\cdots\!28\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(47\!\cdots\!37\)\( T^{16} - \)\(18\!\cdots\!22\)\( T^{17} + \)\(47\!\cdots\!37\)\( p^{3} T^{18} - \)\(29\!\cdots\!40\)\( p^{6} T^{19} + \)\(85\!\cdots\!28\)\( p^{9} T^{20} - \)\(26\!\cdots\!96\)\( p^{12} T^{21} + \)\(13\!\cdots\!78\)\( p^{15} T^{22} + \)\(25\!\cdots\!56\)\( p^{18} T^{23} + \)\(18\!\cdots\!32\)\( p^{21} T^{24} + \)\(48\!\cdots\!90\)\( p^{24} T^{25} + \)\(20\!\cdots\!43\)\( p^{27} T^{26} + 799167654299434508 p^{30} T^{27} + 18048779405752596 p^{33} T^{28} + 7212382055440 p^{36} T^{29} + 115681778818 p^{39} T^{30} + 37370988 p^{42} T^{31} + 484512 p^{45} T^{32} + 88 p^{48} T^{33} + p^{51} T^{34} \) | |
| 41 | \( 1 - 512 T + 913001 T^{2} - 389409122 T^{3} + 389501803079 T^{4} - 142092248300122 T^{5} + 104308712019758420 T^{6} - 33094411000475185402 T^{7} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(55\!\cdots\!10\)\( T^{9} + \)\(28\!\cdots\!85\)\( T^{10} - \)\(71\!\cdots\!80\)\( T^{11} + \)\(33\!\cdots\!47\)\( T^{12} - \)\(73\!\cdots\!72\)\( T^{13} + \)\(31\!\cdots\!49\)\( T^{14} - \)\(63\!\cdots\!80\)\( T^{15} + \)\(25\!\cdots\!14\)\( T^{16} - \)\(47\!\cdots\!66\)\( T^{17} + \)\(25\!\cdots\!14\)\( p^{3} T^{18} - \)\(63\!\cdots\!80\)\( p^{6} T^{19} + \)\(31\!\cdots\!49\)\( p^{9} T^{20} - \)\(73\!\cdots\!72\)\( p^{12} T^{21} + \)\(33\!\cdots\!47\)\( p^{15} T^{22} - \)\(71\!\cdots\!80\)\( p^{18} T^{23} + \)\(28\!\cdots\!85\)\( p^{21} T^{24} - \)\(55\!\cdots\!10\)\( p^{24} T^{25} + \)\(19\!\cdots\!11\)\( p^{27} T^{26} - 33094411000475185402 p^{30} T^{27} + 104308712019758420 p^{33} T^{28} - 142092248300122 p^{36} T^{29} + 389501803079 p^{39} T^{30} - 389409122 p^{42} T^{31} + 913001 p^{45} T^{32} - 512 p^{48} T^{33} + p^{51} T^{34} \) | |
| 43 | \( 1 - 927 T + 1033091 T^{2} - 695542952 T^{3} + 477409316155 T^{4} - 259643809803275 T^{5} + 139018209210306198 T^{6} - 64622278529701544213 T^{7} + \)\(29\!\cdots\!27\)\( T^{8} - \)\(12\!\cdots\!36\)\( T^{9} + \)\(48\!\cdots\!17\)\( T^{10} - \)\(17\!\cdots\!57\)\( T^{11} + \)\(64\!\cdots\!48\)\( T^{12} - \)\(21\!\cdots\!53\)\( T^{13} + \)\(71\!\cdots\!81\)\( T^{14} - \)\(22\!\cdots\!22\)\( T^{15} + \)\(66\!\cdots\!24\)\( T^{16} - \)\(19\!\cdots\!94\)\( T^{17} + \)\(66\!\cdots\!24\)\( p^{3} T^{18} - \)\(22\!\cdots\!22\)\( p^{6} T^{19} + \)\(71\!\cdots\!81\)\( p^{9} T^{20} - \)\(21\!\cdots\!53\)\( p^{12} T^{21} + \)\(64\!\cdots\!48\)\( p^{15} T^{22} - \)\(17\!\cdots\!57\)\( p^{18} T^{23} + \)\(48\!\cdots\!17\)\( p^{21} T^{24} - \)\(12\!\cdots\!36\)\( p^{24} T^{25} + \)\(29\!\cdots\!27\)\( p^{27} T^{26} - 64622278529701544213 p^{30} T^{27} + 139018209210306198 p^{33} T^{28} - 259643809803275 p^{36} T^{29} + 477409316155 p^{39} T^{30} - 695542952 p^{42} T^{31} + 1033091 p^{45} T^{32} - 927 p^{48} T^{33} + p^{51} T^{34} \) | |
| 47 | \( 1 - 143 T + 891461 T^{2} - 92586132 T^{3} + 408188667955 T^{4} - 24596699950109 T^{5} + 125386324853001294 T^{6} - 1998134235622418665 T^{7} + \)\(28\!\cdots\!33\)\( T^{8} + \)\(78\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!91\)\( T^{10} + \)\(35\!\cdots\!31\)\( T^{11} + \)\(82\!\cdots\!52\)\( T^{12} + \)\(78\!\cdots\!57\)\( T^{13} + \)\(11\!\cdots\!77\)\( T^{14} + \)\(12\!\cdots\!50\)\( T^{15} + \)\(12\!\cdots\!34\)\( T^{16} + \)\(14\!\cdots\!90\)\( T^{17} + \)\(12\!\cdots\!34\)\( p^{3} T^{18} + \)\(12\!\cdots\!50\)\( p^{6} T^{19} + \)\(11\!\cdots\!77\)\( p^{9} T^{20} + \)\(78\!\cdots\!57\)\( p^{12} T^{21} + \)\(82\!\cdots\!52\)\( p^{15} T^{22} + \)\(35\!\cdots\!31\)\( p^{18} T^{23} + \)\(53\!\cdots\!91\)\( p^{21} T^{24} + \)\(78\!\cdots\!00\)\( p^{24} T^{25} + \)\(28\!\cdots\!33\)\( p^{27} T^{26} - 1998134235622418665 p^{30} T^{27} + 125386324853001294 p^{33} T^{28} - 24596699950109 p^{36} T^{29} + 408188667955 p^{39} T^{30} - 92586132 p^{42} T^{31} + 891461 p^{45} T^{32} - 143 p^{48} T^{33} + p^{51} T^{34} \) | |
| 53 | \( 1 - 2 p T + 1388431 T^{2} - 69581194 T^{3} + 936596881995 T^{4} + 1650229409704 T^{5} + 409613909279347398 T^{6} + 22491205873430784566 T^{7} + \)\(13\!\cdots\!25\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(32\!\cdots\!69\)\( T^{10} + \)\(57\!\cdots\!72\)\( T^{11} + \)\(65\!\cdots\!05\)\( T^{12} + \)\(15\!\cdots\!12\)\( T^{13} + \)\(11\!\cdots\!91\)\( T^{14} + \)\(33\!\cdots\!04\)\( T^{15} + \)\(18\!\cdots\!94\)\( T^{16} + \)\(56\!\cdots\!82\)\( T^{17} + \)\(18\!\cdots\!94\)\( p^{3} T^{18} + \)\(33\!\cdots\!04\)\( p^{6} T^{19} + \)\(11\!\cdots\!91\)\( p^{9} T^{20} + \)\(15\!\cdots\!12\)\( p^{12} T^{21} + \)\(65\!\cdots\!05\)\( p^{15} T^{22} + \)\(57\!\cdots\!72\)\( p^{18} T^{23} + \)\(32\!\cdots\!69\)\( p^{21} T^{24} + \)\(14\!\cdots\!80\)\( p^{24} T^{25} + \)\(13\!\cdots\!25\)\( p^{27} T^{26} + 22491205873430784566 p^{30} T^{27} + 409613909279347398 p^{33} T^{28} + 1650229409704 p^{36} T^{29} + 936596881995 p^{39} T^{30} - 69581194 p^{42} T^{31} + 1388431 p^{45} T^{32} - 2 p^{49} T^{33} + p^{51} T^{34} \) | |
| 59 | \( 1 - 266 T + 1746671 T^{2} - 455903074 T^{3} + 1515852229255 T^{4} - 356191585936238 T^{5} + 861421856537773846 T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(35\!\cdots\!83\)\( T^{8} - \)\(46\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} - \)\(61\!\cdots\!12\)\( T^{11} + \)\(29\!\cdots\!88\)\( T^{12} + \)\(10\!\cdots\!98\)\( T^{13} + \)\(65\!\cdots\!69\)\( T^{14} + \)\(78\!\cdots\!50\)\( T^{15} + \)\(13\!\cdots\!40\)\( T^{16} + \)\(21\!\cdots\!80\)\( T^{17} + \)\(13\!\cdots\!40\)\( p^{3} T^{18} + \)\(78\!\cdots\!50\)\( p^{6} T^{19} + \)\(65\!\cdots\!69\)\( p^{9} T^{20} + \)\(10\!\cdots\!98\)\( p^{12} T^{21} + \)\(29\!\cdots\!88\)\( p^{15} T^{22} - \)\(61\!\cdots\!12\)\( p^{18} T^{23} + \)\(11\!\cdots\!49\)\( p^{21} T^{24} - \)\(46\!\cdots\!96\)\( p^{24} T^{25} + \)\(35\!\cdots\!83\)\( p^{27} T^{26} - \)\(16\!\cdots\!92\)\( p^{30} T^{27} + 861421856537773846 p^{33} T^{28} - 356191585936238 p^{36} T^{29} + 1515852229255 p^{39} T^{30} - 455903074 p^{42} T^{31} + 1746671 p^{45} T^{32} - 266 p^{48} T^{33} + p^{51} T^{34} \) | |
| 61 | \( 1 + 624 T + 2092824 T^{2} + 1256440162 T^{3} + 2276923441258 T^{4} + 1291961401866140 T^{5} + 1686435943199576446 T^{6} + \)\(89\!\cdots\!20\)\( T^{7} + \)\(94\!\cdots\!39\)\( T^{8} + \)\(47\!\cdots\!10\)\( T^{9} + \)\(42\!\cdots\!36\)\( T^{10} + \)\(19\!\cdots\!58\)\( T^{11} + \)\(15\!\cdots\!72\)\( T^{12} + \)\(68\!\cdots\!88\)\( T^{13} + \)\(49\!\cdots\!82\)\( T^{14} + \)\(20\!\cdots\!38\)\( T^{15} + \)\(13\!\cdots\!73\)\( T^{16} + \)\(49\!\cdots\!98\)\( T^{17} + \)\(13\!\cdots\!73\)\( p^{3} T^{18} + \)\(20\!\cdots\!38\)\( p^{6} T^{19} + \)\(49\!\cdots\!82\)\( p^{9} T^{20} + \)\(68\!\cdots\!88\)\( p^{12} T^{21} + \)\(15\!\cdots\!72\)\( p^{15} T^{22} + \)\(19\!\cdots\!58\)\( p^{18} T^{23} + \)\(42\!\cdots\!36\)\( p^{21} T^{24} + \)\(47\!\cdots\!10\)\( p^{24} T^{25} + \)\(94\!\cdots\!39\)\( p^{27} T^{26} + \)\(89\!\cdots\!20\)\( p^{30} T^{27} + 1686435943199576446 p^{33} T^{28} + 1291961401866140 p^{36} T^{29} + 2276923441258 p^{39} T^{30} + 1256440162 p^{42} T^{31} + 2092824 p^{45} T^{32} + 624 p^{48} T^{33} + p^{51} T^{34} \) | |
| 67 | \( 1 - 676 T + 1705634 T^{2} - 744454014 T^{3} + 1380574893067 T^{4} - 358836095300068 T^{5} + 733672028479072624 T^{6} - 44421690747321027402 T^{7} + \)\(28\!\cdots\!24\)\( T^{8} + \)\(54\!\cdots\!00\)\( T^{9} + \)\(96\!\cdots\!10\)\( T^{10} + \)\(43\!\cdots\!86\)\( T^{11} + \)\(33\!\cdots\!46\)\( T^{12} + \)\(19\!\cdots\!28\)\( T^{13} + \)\(12\!\cdots\!93\)\( T^{14} + \)\(65\!\cdots\!86\)\( T^{15} + \)\(43\!\cdots\!95\)\( T^{16} + \)\(19\!\cdots\!72\)\( T^{17} + \)\(43\!\cdots\!95\)\( p^{3} T^{18} + \)\(65\!\cdots\!86\)\( p^{6} T^{19} + \)\(12\!\cdots\!93\)\( p^{9} T^{20} + \)\(19\!\cdots\!28\)\( p^{12} T^{21} + \)\(33\!\cdots\!46\)\( p^{15} T^{22} + \)\(43\!\cdots\!86\)\( p^{18} T^{23} + \)\(96\!\cdots\!10\)\( p^{21} T^{24} + \)\(54\!\cdots\!00\)\( p^{24} T^{25} + \)\(28\!\cdots\!24\)\( p^{27} T^{26} - 44421690747321027402 p^{30} T^{27} + 733672028479072624 p^{33} T^{28} - 358836095300068 p^{36} T^{29} + 1380574893067 p^{39} T^{30} - 744454014 p^{42} T^{31} + 1705634 p^{45} T^{32} - 676 p^{48} T^{33} + p^{51} T^{34} \) | |
| 71 | \( 1 - 763 T + 4039146 T^{2} - 3207411103 T^{3} + 8065900264589 T^{4} - 6467705289181054 T^{5} + 10595356578758633370 T^{6} - \)\(83\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!92\)\( T^{8} - \)\(77\!\cdots\!58\)\( T^{9} + \)\(77\!\cdots\!58\)\( T^{10} - \)\(55\!\cdots\!86\)\( T^{11} + \)\(47\!\cdots\!16\)\( T^{12} - \)\(31\!\cdots\!98\)\( T^{13} + \)\(24\!\cdots\!67\)\( T^{14} - \)\(20\!\cdots\!33\)\( p T^{15} + \)\(10\!\cdots\!63\)\( T^{16} - \)\(58\!\cdots\!54\)\( T^{17} + \)\(10\!\cdots\!63\)\( p^{3} T^{18} - \)\(20\!\cdots\!33\)\( p^{7} T^{19} + \)\(24\!\cdots\!67\)\( p^{9} T^{20} - \)\(31\!\cdots\!98\)\( p^{12} T^{21} + \)\(47\!\cdots\!16\)\( p^{15} T^{22} - \)\(55\!\cdots\!86\)\( p^{18} T^{23} + \)\(77\!\cdots\!58\)\( p^{21} T^{24} - \)\(77\!\cdots\!58\)\( p^{24} T^{25} + \)\(10\!\cdots\!92\)\( p^{27} T^{26} - \)\(83\!\cdots\!00\)\( p^{30} T^{27} + 10595356578758633370 p^{33} T^{28} - 6467705289181054 p^{36} T^{29} + 8065900264589 p^{39} T^{30} - 3207411103 p^{42} T^{31} + 4039146 p^{45} T^{32} - 763 p^{48} T^{33} + p^{51} T^{34} \) | |
| 73 | \( 1 - 2374 T + 5743838 T^{2} - 8886707052 T^{3} + 13203045640021 T^{4} - 15722706189915188 T^{5} + 17918778634302008141 T^{6} - \)\(17\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!61\)\( T^{10} - \)\(90\!\cdots\!44\)\( T^{11} + \)\(67\!\cdots\!38\)\( T^{12} - \)\(46\!\cdots\!88\)\( T^{13} + \)\(32\!\cdots\!44\)\( T^{14} - \)\(20\!\cdots\!28\)\( T^{15} + \)\(13\!\cdots\!26\)\( T^{16} - \)\(84\!\cdots\!20\)\( T^{17} + \)\(13\!\cdots\!26\)\( p^{3} T^{18} - \)\(20\!\cdots\!28\)\( p^{6} T^{19} + \)\(32\!\cdots\!44\)\( p^{9} T^{20} - \)\(46\!\cdots\!88\)\( p^{12} T^{21} + \)\(67\!\cdots\!38\)\( p^{15} T^{22} - \)\(90\!\cdots\!44\)\( p^{18} T^{23} + \)\(11\!\cdots\!61\)\( p^{21} T^{24} - \)\(14\!\cdots\!50\)\( p^{24} T^{25} + \)\(16\!\cdots\!22\)\( p^{27} T^{26} - \)\(17\!\cdots\!60\)\( p^{30} T^{27} + 17918778634302008141 p^{33} T^{28} - 15722706189915188 p^{36} T^{29} + 13203045640021 p^{39} T^{30} - 8886707052 p^{42} T^{31} + 5743838 p^{45} T^{32} - 2374 p^{48} T^{33} + p^{51} T^{34} \) | |
| 79 | \( 1 - 2164 T + 6332579 T^{2} - 10493988684 T^{3} + 18460433904316 T^{4} - 24975923474321068 T^{5} + 33711726912150524960 T^{6} - \)\(38\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!36\)\( T^{8} - \)\(44\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!60\)\( T^{10} - \)\(39\!\cdots\!64\)\( T^{11} + \)\(34\!\cdots\!08\)\( T^{12} - \)\(28\!\cdots\!12\)\( T^{13} + \)\(29\!\cdots\!36\)\( p T^{14} - \)\(17\!\cdots\!40\)\( T^{15} + \)\(13\!\cdots\!94\)\( T^{16} - \)\(93\!\cdots\!44\)\( T^{17} + \)\(13\!\cdots\!94\)\( p^{3} T^{18} - \)\(17\!\cdots\!40\)\( p^{6} T^{19} + \)\(29\!\cdots\!36\)\( p^{10} T^{20} - \)\(28\!\cdots\!12\)\( p^{12} T^{21} + \)\(34\!\cdots\!08\)\( p^{15} T^{22} - \)\(39\!\cdots\!64\)\( p^{18} T^{23} + \)\(43\!\cdots\!60\)\( p^{21} T^{24} - \)\(44\!\cdots\!76\)\( p^{24} T^{25} + \)\(43\!\cdots\!36\)\( p^{27} T^{26} - \)\(38\!\cdots\!40\)\( p^{30} T^{27} + 33711726912150524960 p^{33} T^{28} - 24975923474321068 p^{36} T^{29} + 18460433904316 p^{39} T^{30} - 10493988684 p^{42} T^{31} + 6332579 p^{45} T^{32} - 2164 p^{48} T^{33} + p^{51} T^{34} \) | |
| 83 | \( 1 + 777 T + 5295175 T^{2} + 4289092528 T^{3} + 14597990353951 T^{4} + 11878072760855465 T^{5} + 27524805544257304598 T^{6} + \)\(21\!\cdots\!39\)\( T^{7} + \)\(39\!\cdots\!47\)\( T^{8} + \)\(30\!\cdots\!20\)\( T^{9} + \)\(45\!\cdots\!73\)\( T^{10} + \)\(32\!\cdots\!03\)\( T^{11} + \)\(42\!\cdots\!00\)\( T^{12} + \)\(29\!\cdots\!83\)\( T^{13} + \)\(33\!\cdots\!09\)\( T^{14} + \)\(21\!\cdots\!30\)\( T^{15} + \)\(22\!\cdots\!60\)\( T^{16} + \)\(13\!\cdots\!90\)\( T^{17} + \)\(22\!\cdots\!60\)\( p^{3} T^{18} + \)\(21\!\cdots\!30\)\( p^{6} T^{19} + \)\(33\!\cdots\!09\)\( p^{9} T^{20} + \)\(29\!\cdots\!83\)\( p^{12} T^{21} + \)\(42\!\cdots\!00\)\( p^{15} T^{22} + \)\(32\!\cdots\!03\)\( p^{18} T^{23} + \)\(45\!\cdots\!73\)\( p^{21} T^{24} + \)\(30\!\cdots\!20\)\( p^{24} T^{25} + \)\(39\!\cdots\!47\)\( p^{27} T^{26} + \)\(21\!\cdots\!39\)\( p^{30} T^{27} + 27524805544257304598 p^{33} T^{28} + 11878072760855465 p^{36} T^{29} + 14597990353951 p^{39} T^{30} + 4289092528 p^{42} T^{31} + 5295175 p^{45} T^{32} + 777 p^{48} T^{33} + p^{51} T^{34} \) | |
| 89 | \( 1 - 1687 T + 4676684 T^{2} - 5442682587 T^{3} + 10192816098481 T^{4} - 9421379650114436 T^{5} + 14857209915663399912 T^{6} - \)\(11\!\cdots\!30\)\( T^{7} + \)\(17\!\cdots\!16\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{9} + \)\(18\!\cdots\!82\)\( T^{10} - \)\(12\!\cdots\!12\)\( T^{11} + \)\(17\!\cdots\!38\)\( T^{12} - \)\(10\!\cdots\!56\)\( T^{13} + \)\(15\!\cdots\!33\)\( T^{14} - \)\(84\!\cdots\!07\)\( T^{15} + \)\(11\!\cdots\!13\)\( T^{16} - \)\(61\!\cdots\!06\)\( T^{17} + \)\(11\!\cdots\!13\)\( p^{3} T^{18} - \)\(84\!\cdots\!07\)\( p^{6} T^{19} + \)\(15\!\cdots\!33\)\( p^{9} T^{20} - \)\(10\!\cdots\!56\)\( p^{12} T^{21} + \)\(17\!\cdots\!38\)\( p^{15} T^{22} - \)\(12\!\cdots\!12\)\( p^{18} T^{23} + \)\(18\!\cdots\!82\)\( p^{21} T^{24} - \)\(12\!\cdots\!70\)\( p^{24} T^{25} + \)\(17\!\cdots\!16\)\( p^{27} T^{26} - \)\(11\!\cdots\!30\)\( p^{30} T^{27} + 14857209915663399912 p^{33} T^{28} - 9421379650114436 p^{36} T^{29} + 10192816098481 p^{39} T^{30} - 5442682587 p^{42} T^{31} + 4676684 p^{45} T^{32} - 1687 p^{48} T^{33} + p^{51} T^{34} \) | |
| 97 | \( 1 - 2047 T + 10794246 T^{2} - 16475427899 T^{3} + 49852750799026 T^{4} - 58378719187539875 T^{5} + \)\(13\!\cdots\!07\)\( T^{6} - \)\(12\!\cdots\!70\)\( T^{7} + \)\(24\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!56\)\( T^{9} + \)\(33\!\cdots\!76\)\( T^{10} - \)\(16\!\cdots\!80\)\( T^{11} + \)\(41\!\cdots\!31\)\( T^{12} - \)\(16\!\cdots\!53\)\( T^{13} + \)\(48\!\cdots\!68\)\( T^{14} - \)\(18\!\cdots\!55\)\( T^{15} + \)\(51\!\cdots\!07\)\( T^{16} - \)\(18\!\cdots\!62\)\( T^{17} + \)\(51\!\cdots\!07\)\( p^{3} T^{18} - \)\(18\!\cdots\!55\)\( p^{6} T^{19} + \)\(48\!\cdots\!68\)\( p^{9} T^{20} - \)\(16\!\cdots\!53\)\( p^{12} T^{21} + \)\(41\!\cdots\!31\)\( p^{15} T^{22} - \)\(16\!\cdots\!80\)\( p^{18} T^{23} + \)\(33\!\cdots\!76\)\( p^{21} T^{24} - \)\(16\!\cdots\!56\)\( p^{24} T^{25} + \)\(24\!\cdots\!02\)\( p^{27} T^{26} - \)\(12\!\cdots\!70\)\( p^{30} T^{27} + \)\(13\!\cdots\!07\)\( p^{33} T^{28} - 58378719187539875 p^{36} T^{29} + 49852750799026 p^{39} T^{30} - 16475427899 p^{42} T^{31} + 10794246 p^{45} T^{32} - 2047 p^{48} T^{33} + p^{51} T^{34} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.84951387046190760881716882158, −1.81108794704202182908869034825, −1.63904628845948478822496721122, −1.51640666951415251911315862616, −1.48530719832273130202472320697, −1.47043360430809870312168163600, −1.42473515074620474257360068560, −1.33542630170129152520547781194, −1.23933955005970867389491257830, −1.23550140470815638626034299897, −0.889881243933661333649417501636, −0.886884296399652675464104149681, −0.819445097115968707371514779470, −0.74650688427938687889606029503, −0.64952427275426442178416872862, −0.63125551096360009207781495939, −0.50962959652622752803937041324, −0.48517167702281484941598005853, −0.40200275980941054289822525968, −0.38866408687409483617718916760, −0.36122549555068755171354493215, −0.22042508724246486479279787372, −0.18150503940851340304253975605, −0.081361087273186869222923178794, −0.02194761718014676586745034618, 0.02194761718014676586745034618, 0.081361087273186869222923178794, 0.18150503940851340304253975605, 0.22042508724246486479279787372, 0.36122549555068755171354493215, 0.38866408687409483617718916760, 0.40200275980941054289822525968, 0.48517167702281484941598005853, 0.50962959652622752803937041324, 0.63125551096360009207781495939, 0.64952427275426442178416872862, 0.74650688427938687889606029503, 0.819445097115968707371514779470, 0.886884296399652675464104149681, 0.889881243933661333649417501636, 1.23550140470815638626034299897, 1.23933955005970867389491257830, 1.33542630170129152520547781194, 1.42473515074620474257360068560, 1.47043360430809870312168163600, 1.48530719832273130202472320697, 1.51640666951415251911315862616, 1.63904628845948478822496721122, 1.81108794704202182908869034825, 1.84951387046190760881716882158
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.