Dirichlet series
| L(s) = 1 | − 2·2-s + 7·3-s − 10·4-s − 14·6-s + 3·7-s + 23·8-s + 5·9-s − 10·11-s − 70·12-s + 8·13-s − 6·14-s + 41·16-s + 17-s − 10·18-s − 30·19-s + 21·21-s + 20·22-s − 19·23-s + 161·24-s − 16·26-s − 79·27-s − 30·28-s − 12·29-s − 22·31-s − 116·32-s − 70·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4.04·3-s − 5·4-s − 5.71·6-s + 1.13·7-s + 8.13·8-s + 5/3·9-s − 3.01·11-s − 20.2·12-s + 2.21·13-s − 1.60·14-s + 41/4·16-s + 0.242·17-s − 2.35·18-s − 6.88·19-s + 4.58·21-s + 4.26·22-s − 3.96·23-s + 32.8·24-s − 3.13·26-s − 15.2·27-s − 5.66·28-s − 2.22·29-s − 3.95·31-s − 20.5·32-s − 12.1·33-s − 0.342·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{30} \cdot 241^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{30} \cdot 241^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(5^{30} \cdot 241^{15}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.71203\times 10^{25}\) |
| Root analytic conductor: | \(6.93612\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(15\) |
| Selberg data: | \((30,\ 5^{30} \cdot 241^{15} ,\ ( \ : [1/2]^{15} ),\ -1 )\) |
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 | 5 | \( 1 \) |
| 241 | \( ( 1 + T )^{15} \) | |
| good | 2 | \( 1 + p T + 7 p T^{2} + 25 T^{3} + 103 T^{4} + 21 p^{3} T^{5} + 265 p T^{6} + 801 T^{7} + 2121 T^{8} + 2987 T^{9} + 6933 T^{10} + 4557 p T^{11} + 19011 T^{12} + 23293 T^{13} + 44373 T^{14} + 50485 T^{15} + 44373 p T^{16} + 23293 p^{2} T^{17} + 19011 p^{3} T^{18} + 4557 p^{5} T^{19} + 6933 p^{5} T^{20} + 2987 p^{6} T^{21} + 2121 p^{7} T^{22} + 801 p^{8} T^{23} + 265 p^{10} T^{24} + 21 p^{13} T^{25} + 103 p^{11} T^{26} + 25 p^{12} T^{27} + 7 p^{14} T^{28} + p^{15} T^{29} + p^{15} T^{30} \) |
| 3 | \( 1 - 7 T + 44 T^{2} - 194 T^{3} + 773 T^{4} - 2617 T^{5} + 8164 T^{6} - 22967 T^{7} + 60449 T^{8} - 147310 T^{9} + 339467 T^{10} - 734507 T^{11} + 1513337 T^{12} - 2950637 T^{13} + 1833272 p T^{14} - 9736601 T^{15} + 1833272 p^{2} T^{16} - 2950637 p^{2} T^{17} + 1513337 p^{3} T^{18} - 734507 p^{4} T^{19} + 339467 p^{5} T^{20} - 147310 p^{6} T^{21} + 60449 p^{7} T^{22} - 22967 p^{8} T^{23} + 8164 p^{9} T^{24} - 2617 p^{10} T^{25} + 773 p^{11} T^{26} - 194 p^{12} T^{27} + 44 p^{13} T^{28} - 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 - 3 T + 69 T^{2} - 184 T^{3} + 2285 T^{4} - 5435 T^{5} + 48514 T^{6} - 103450 T^{7} + 746705 T^{8} - 29329 p^{2} T^{9} + 8946471 T^{10} - 15668832 T^{11} + 12488390 p T^{12} - 140495069 T^{13} + 717670379 T^{14} - 1064822293 T^{15} + 717670379 p T^{16} - 140495069 p^{2} T^{17} + 12488390 p^{4} T^{18} - 15668832 p^{4} T^{19} + 8946471 p^{5} T^{20} - 29329 p^{8} T^{21} + 746705 p^{7} T^{22} - 103450 p^{8} T^{23} + 48514 p^{9} T^{24} - 5435 p^{10} T^{25} + 2285 p^{11} T^{26} - 184 p^{12} T^{27} + 69 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 11 | \( 1 + 10 T + 140 T^{2} + 1028 T^{3} + 8603 T^{4} + 4637 p T^{5} + 324685 T^{6} + 1638350 T^{7} + 8671398 T^{8} + 38435840 T^{9} + 176810244 T^{10} + 701777246 T^{11} + 2875783160 T^{12} + 10334097764 T^{13} + 38247516248 T^{14} + 125038989415 T^{15} + 38247516248 p T^{16} + 10334097764 p^{2} T^{17} + 2875783160 p^{3} T^{18} + 701777246 p^{4} T^{19} + 176810244 p^{5} T^{20} + 38435840 p^{6} T^{21} + 8671398 p^{7} T^{22} + 1638350 p^{8} T^{23} + 324685 p^{9} T^{24} + 4637 p^{11} T^{25} + 8603 p^{11} T^{26} + 1028 p^{12} T^{27} + 140 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 - 8 T + 135 T^{2} - 856 T^{3} + 626 p T^{4} - 42812 T^{5} + 297820 T^{6} - 1347080 T^{7} + 7616486 T^{8} - 30614433 T^{9} + 149729256 T^{10} - 552303058 T^{11} + 2443009380 T^{12} - 8488918786 T^{13} + 35043186281 T^{14} - 116137458321 T^{15} + 35043186281 p T^{16} - 8488918786 p^{2} T^{17} + 2443009380 p^{3} T^{18} - 552303058 p^{4} T^{19} + 149729256 p^{5} T^{20} - 30614433 p^{6} T^{21} + 7616486 p^{7} T^{22} - 1347080 p^{8} T^{23} + 297820 p^{9} T^{24} - 42812 p^{10} T^{25} + 626 p^{12} T^{26} - 856 p^{12} T^{27} + 135 p^{13} T^{28} - 8 p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 - T + 126 T^{2} - 269 T^{3} + 8190 T^{4} - 24298 T^{5} + 373120 T^{6} - 1283146 T^{7} + 13229934 T^{8} - 48031043 T^{9} + 380644474 T^{10} - 1384341208 T^{11} + 9095231436 T^{12} - 31895449037 T^{13} + 182878337902 T^{14} - 598744414275 T^{15} + 182878337902 p T^{16} - 31895449037 p^{2} T^{17} + 9095231436 p^{3} T^{18} - 1384341208 p^{4} T^{19} + 380644474 p^{5} T^{20} - 48031043 p^{6} T^{21} + 13229934 p^{7} T^{22} - 1283146 p^{8} T^{23} + 373120 p^{9} T^{24} - 24298 p^{10} T^{25} + 8190 p^{11} T^{26} - 269 p^{12} T^{27} + 126 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 + 30 T + 617 T^{2} + 9225 T^{3} + 114598 T^{4} + 1205122 T^{5} + 11195374 T^{6} + 92718239 T^{7} + 698401371 T^{8} + 4807897553 T^{9} + 30589810478 T^{10} + 9488968460 p T^{11} + 991023236386 T^{12} + 5083321480560 T^{13} + 24432500370967 T^{14} + 109938656945393 T^{15} + 24432500370967 p T^{16} + 5083321480560 p^{2} T^{17} + 991023236386 p^{3} T^{18} + 9488968460 p^{5} T^{19} + 30589810478 p^{5} T^{20} + 4807897553 p^{6} T^{21} + 698401371 p^{7} T^{22} + 92718239 p^{8} T^{23} + 11195374 p^{9} T^{24} + 1205122 p^{10} T^{25} + 114598 p^{11} T^{26} + 9225 p^{12} T^{27} + 617 p^{13} T^{28} + 30 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 + 19 T + 15 p T^{2} + 4344 T^{3} + 48872 T^{4} + 467208 T^{5} + 4038140 T^{6} + 31447888 T^{7} + 225466890 T^{8} + 1495229706 T^{9} + 9276905985 T^{10} + 54168988017 T^{11} + 300138423930 T^{12} + 1586908275369 T^{13} + 8051845774445 T^{14} + 39328563854155 T^{15} + 8051845774445 p T^{16} + 1586908275369 p^{2} T^{17} + 300138423930 p^{3} T^{18} + 54168988017 p^{4} T^{19} + 9276905985 p^{5} T^{20} + 1495229706 p^{6} T^{21} + 225466890 p^{7} T^{22} + 31447888 p^{8} T^{23} + 4038140 p^{9} T^{24} + 467208 p^{10} T^{25} + 48872 p^{11} T^{26} + 4344 p^{12} T^{27} + 15 p^{14} T^{28} + 19 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 + 12 T + 339 T^{2} + 3188 T^{3} + 50639 T^{4} + 382964 T^{5} + 4444766 T^{6} + 27152996 T^{7} + 256269972 T^{8} + 1242735270 T^{9} + 10314402824 T^{10} + 38009283932 T^{11} + 309955761183 T^{12} + 831254958912 T^{13} + 8114646936272 T^{14} + 18821183700381 T^{15} + 8114646936272 p T^{16} + 831254958912 p^{2} T^{17} + 309955761183 p^{3} T^{18} + 38009283932 p^{4} T^{19} + 10314402824 p^{5} T^{20} + 1242735270 p^{6} T^{21} + 256269972 p^{7} T^{22} + 27152996 p^{8} T^{23} + 4444766 p^{9} T^{24} + 382964 p^{10} T^{25} + 50639 p^{11} T^{26} + 3188 p^{12} T^{27} + 339 p^{13} T^{28} + 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + 22 T + 440 T^{2} + 5691 T^{3} + 69881 T^{4} + 688745 T^{5} + 6648869 T^{6} + 55558941 T^{7} + 460139509 T^{8} + 3403260763 T^{9} + 25026167067 T^{10} + 166971287488 T^{11} + 1109762228609 T^{12} + 6751797338530 T^{13} + 40964110917559 T^{14} + 228369831804309 T^{15} + 40964110917559 p T^{16} + 6751797338530 p^{2} T^{17} + 1109762228609 p^{3} T^{18} + 166971287488 p^{4} T^{19} + 25026167067 p^{5} T^{20} + 3403260763 p^{6} T^{21} + 460139509 p^{7} T^{22} + 55558941 p^{8} T^{23} + 6648869 p^{9} T^{24} + 688745 p^{10} T^{25} + 69881 p^{11} T^{26} + 5691 p^{12} T^{27} + 440 p^{13} T^{28} + 22 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 - 12 T + 314 T^{2} - 3412 T^{3} + 1354 p T^{4} - 494865 T^{5} + 5446581 T^{6} - 48638912 T^{7} + 448085578 T^{8} - 3616250406 T^{9} + 29279798873 T^{10} - 214642130278 T^{11} + 1561641890826 T^{12} - 10455854594328 T^{13} + 69101820742268 T^{14} - 423482062973499 T^{15} + 69101820742268 p T^{16} - 10455854594328 p^{2} T^{17} + 1561641890826 p^{3} T^{18} - 214642130278 p^{4} T^{19} + 29279798873 p^{5} T^{20} - 3616250406 p^{6} T^{21} + 448085578 p^{7} T^{22} - 48638912 p^{8} T^{23} + 5446581 p^{9} T^{24} - 494865 p^{10} T^{25} + 1354 p^{12} T^{26} - 3412 p^{12} T^{27} + 314 p^{13} T^{28} - 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 13 T + 341 T^{2} + 3629 T^{3} + 52543 T^{4} + 466994 T^{5} + 4839600 T^{6} + 36354346 T^{7} + 298170361 T^{8} + 1902608617 T^{9} + 13180820992 T^{10} + 71589476876 T^{11} + 455476595943 T^{12} + 2172102398188 T^{13} + 14836124371336 T^{14} + 73017536046731 T^{15} + 14836124371336 p T^{16} + 2172102398188 p^{2} T^{17} + 455476595943 p^{3} T^{18} + 71589476876 p^{4} T^{19} + 13180820992 p^{5} T^{20} + 1902608617 p^{6} T^{21} + 298170361 p^{7} T^{22} + 36354346 p^{8} T^{23} + 4839600 p^{9} T^{24} + 466994 p^{10} T^{25} + 52543 p^{11} T^{26} + 3629 p^{12} T^{27} + 341 p^{13} T^{28} + 13 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 - 25 T + 664 T^{2} - 11070 T^{3} + 178891 T^{4} - 2300836 T^{5} + 28352101 T^{6} - 301544277 T^{7} + 3080085239 T^{8} - 28211205837 T^{9} + 249502562636 T^{10} - 2021210853870 T^{11} + 15897696311733 T^{12} - 115971363872945 T^{13} + 824791787040357 T^{14} - 5473357413730351 T^{15} + 824791787040357 p T^{16} - 115971363872945 p^{2} T^{17} + 15897696311733 p^{3} T^{18} - 2021210853870 p^{4} T^{19} + 249502562636 p^{5} T^{20} - 28211205837 p^{6} T^{21} + 3080085239 p^{7} T^{22} - 301544277 p^{8} T^{23} + 28352101 p^{9} T^{24} - 2300836 p^{10} T^{25} + 178891 p^{11} T^{26} - 11070 p^{12} T^{27} + 664 p^{13} T^{28} - 25 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + 16 T + 492 T^{2} + 5699 T^{3} + 99873 T^{4} + 871873 T^{5} + 11279555 T^{6} + 72810887 T^{7} + 783290515 T^{8} + 3220474720 T^{9} + 33378181673 T^{10} + 17288882580 T^{11} + 707170695241 T^{12} - 7584884037603 T^{13} - 4437797603730 T^{14} - 537960786887469 T^{15} - 4437797603730 p T^{16} - 7584884037603 p^{2} T^{17} + 707170695241 p^{3} T^{18} + 17288882580 p^{4} T^{19} + 33378181673 p^{5} T^{20} + 3220474720 p^{6} T^{21} + 783290515 p^{7} T^{22} + 72810887 p^{8} T^{23} + 11279555 p^{9} T^{24} + 871873 p^{10} T^{25} + 99873 p^{11} T^{26} + 5699 p^{12} T^{27} + 492 p^{13} T^{28} + 16 p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 - 4 T + 464 T^{2} - 1139 T^{3} + 104229 T^{4} - 117368 T^{5} + 288234 p T^{6} + 996280 T^{7} + 1653955482 T^{8} + 1832413844 T^{9} + 141924674600 T^{10} + 273324058399 T^{11} + 10108497538498 T^{12} + 24328445769833 T^{13} + 616167868168061 T^{14} + 1516066725600641 T^{15} + 616167868168061 p T^{16} + 24328445769833 p^{2} T^{17} + 10108497538498 p^{3} T^{18} + 273324058399 p^{4} T^{19} + 141924674600 p^{5} T^{20} + 1832413844 p^{6} T^{21} + 1653955482 p^{7} T^{22} + 996280 p^{8} T^{23} + 288234 p^{10} T^{24} - 117368 p^{10} T^{25} + 104229 p^{11} T^{26} - 1139 p^{12} T^{27} + 464 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 + 50 T + 1656 T^{2} + 41091 T^{3} + 846567 T^{4} + 14990210 T^{5} + 235271476 T^{6} + 3323944602 T^{7} + 42862165094 T^{8} + 508599205506 T^{9} + 5594191512530 T^{10} + 57296999928431 T^{11} + 548677398738234 T^{12} + 4924051700103359 T^{13} + 41498895626633429 T^{14} + 328708825551825739 T^{15} + 41498895626633429 p T^{16} + 4924051700103359 p^{2} T^{17} + 548677398738234 p^{3} T^{18} + 57296999928431 p^{4} T^{19} + 5594191512530 p^{5} T^{20} + 508599205506 p^{6} T^{21} + 42862165094 p^{7} T^{22} + 3323944602 p^{8} T^{23} + 235271476 p^{9} T^{24} + 14990210 p^{10} T^{25} + 846567 p^{11} T^{26} + 41091 p^{12} T^{27} + 1656 p^{13} T^{28} + 50 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 41 T + 1266 T^{2} + 28143 T^{3} + 537421 T^{4} + 8723591 T^{5} + 127846442 T^{6} + 1684406586 T^{7} + 20552628279 T^{8} + 231552538619 T^{9} + 2451586546111 T^{10} + 24315296310024 T^{11} + 228555938207149 T^{12} + 2028097124596243 T^{13} + 17129570660768330 T^{14} + 137038654279098775 T^{15} + 17129570660768330 p T^{16} + 2028097124596243 p^{2} T^{17} + 228555938207149 p^{3} T^{18} + 24315296310024 p^{4} T^{19} + 2451586546111 p^{5} T^{20} + 231552538619 p^{6} T^{21} + 20552628279 p^{7} T^{22} + 1684406586 p^{8} T^{23} + 127846442 p^{9} T^{24} + 8723591 p^{10} T^{25} + 537421 p^{11} T^{26} + 28143 p^{12} T^{27} + 1266 p^{13} T^{28} + 41 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 - 43 T + 1412 T^{2} - 34612 T^{3} + 725254 T^{4} - 13148340 T^{5} + 213596249 T^{6} - 3140878835 T^{7} + 42386968766 T^{8} - 528391319525 T^{9} + 6126124618550 T^{10} - 66308482555692 T^{11} + 672588399620473 T^{12} - 6405417740114407 T^{13} + 57392366612773961 T^{14} - 484028819246150401 T^{15} + 57392366612773961 p T^{16} - 6405417740114407 p^{2} T^{17} + 672588399620473 p^{3} T^{18} - 66308482555692 p^{4} T^{19} + 6126124618550 p^{5} T^{20} - 528391319525 p^{6} T^{21} + 42386968766 p^{7} T^{22} - 3140878835 p^{8} T^{23} + 213596249 p^{9} T^{24} - 13148340 p^{10} T^{25} + 725254 p^{11} T^{26} - 34612 p^{12} T^{27} + 1412 p^{13} T^{28} - 43 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 + 14 T + 714 T^{2} + 9134 T^{3} + 256359 T^{4} + 2978414 T^{5} + 60611852 T^{6} + 640396335 T^{7} + 10494156375 T^{8} + 101038970433 T^{9} + 1405479787448 T^{10} + 12341749470266 T^{11} + 150201975448272 T^{12} + 1201578068635484 T^{13} + 13032406954284030 T^{14} + 94635638066606051 T^{15} + 13032406954284030 p T^{16} + 1201578068635484 p^{2} T^{17} + 150201975448272 p^{3} T^{18} + 12341749470266 p^{4} T^{19} + 1405479787448 p^{5} T^{20} + 101038970433 p^{6} T^{21} + 10494156375 p^{7} T^{22} + 640396335 p^{8} T^{23} + 60611852 p^{9} T^{24} + 2978414 p^{10} T^{25} + 256359 p^{11} T^{26} + 9134 p^{12} T^{27} + 714 p^{13} T^{28} + 14 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 - 10 T + 494 T^{2} - 3751 T^{3} + 122454 T^{4} - 787530 T^{5} + 21586985 T^{6} - 125494338 T^{7} + 3021032393 T^{8} - 16172684699 T^{9} + 348789318111 T^{10} - 1732253844491 T^{11} + 34136454539357 T^{12} - 158366641271012 T^{13} + 2879386341403827 T^{14} - 12464863590107445 T^{15} + 2879386341403827 p T^{16} - 158366641271012 p^{2} T^{17} + 34136454539357 p^{3} T^{18} - 1732253844491 p^{4} T^{19} + 348789318111 p^{5} T^{20} - 16172684699 p^{6} T^{21} + 3021032393 p^{7} T^{22} - 125494338 p^{8} T^{23} + 21586985 p^{9} T^{24} - 787530 p^{10} T^{25} + 122454 p^{11} T^{26} - 3751 p^{12} T^{27} + 494 p^{13} T^{28} - 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 + 44 T + 1760 T^{2} + 47139 T^{3} + 1146715 T^{4} + 22849120 T^{5} + 419964256 T^{6} + 6763660759 T^{7} + 101700966064 T^{8} + 1382466434370 T^{9} + 17690936696558 T^{10} + 208066457594283 T^{11} + 2315894010741594 T^{12} + 23901530554084313 T^{13} + 234140016077896982 T^{14} + 2135073976875200373 T^{15} + 234140016077896982 p T^{16} + 23901530554084313 p^{2} T^{17} + 2315894010741594 p^{3} T^{18} + 208066457594283 p^{4} T^{19} + 17690936696558 p^{5} T^{20} + 1382466434370 p^{6} T^{21} + 101700966064 p^{7} T^{22} + 6763660759 p^{8} T^{23} + 419964256 p^{9} T^{24} + 22849120 p^{10} T^{25} + 1146715 p^{11} T^{26} + 47139 p^{12} T^{27} + 1760 p^{13} T^{28} + 44 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 + 7 T + 555 T^{2} + 4899 T^{3} + 155129 T^{4} + 1499352 T^{5} + 29597218 T^{6} + 276408647 T^{7} + 4274594676 T^{8} + 35670610542 T^{9} + 487071844767 T^{10} + 3584592585017 T^{11} + 45880931425585 T^{12} + 310684877413560 T^{13} + 3884329781286276 T^{14} + 25729978545945169 T^{15} + 3884329781286276 p T^{16} + 310684877413560 p^{2} T^{17} + 45880931425585 p^{3} T^{18} + 3584592585017 p^{4} T^{19} + 487071844767 p^{5} T^{20} + 35670610542 p^{6} T^{21} + 4274594676 p^{7} T^{22} + 276408647 p^{8} T^{23} + 29597218 p^{9} T^{24} + 1499352 p^{10} T^{25} + 155129 p^{11} T^{26} + 4899 p^{12} T^{27} + 555 p^{13} T^{28} + 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 - 4 T + 765 T^{2} - 3907 T^{3} + 286678 T^{4} - 1784983 T^{5} + 70653364 T^{6} - 512993615 T^{7} + 12961873059 T^{8} - 104848697801 T^{9} + 1895706212342 T^{10} - 16279343143646 T^{11} + 230359494603497 T^{12} - 1995366970479321 T^{13} + 23824090802589747 T^{14} - 197140730599035623 T^{15} + 23824090802589747 p T^{16} - 1995366970479321 p^{2} T^{17} + 230359494603497 p^{3} T^{18} - 16279343143646 p^{4} T^{19} + 1895706212342 p^{5} T^{20} - 104848697801 p^{6} T^{21} + 12961873059 p^{7} T^{22} - 512993615 p^{8} T^{23} + 70653364 p^{9} T^{24} - 1784983 p^{10} T^{25} + 286678 p^{11} T^{26} - 3907 p^{12} T^{27} + 765 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 + 9 T + 911 T^{2} + 8593 T^{3} + 418332 T^{4} + 4024990 T^{5} + 127456591 T^{6} + 1224782348 T^{7} + 28721075133 T^{8} + 270358448786 T^{9} + 5062473091192 T^{10} + 45799947473449 T^{11} + 720603157804507 T^{12} + 6142121503270985 T^{13} + 84329245944514849 T^{14} + 662618094351958005 T^{15} + 84329245944514849 p T^{16} + 6142121503270985 p^{2} T^{17} + 720603157804507 p^{3} T^{18} + 45799947473449 p^{4} T^{19} + 5062473091192 p^{5} T^{20} + 270358448786 p^{6} T^{21} + 28721075133 p^{7} T^{22} + 1224782348 p^{8} T^{23} + 127456591 p^{9} T^{24} + 4024990 p^{10} T^{25} + 418332 p^{11} T^{26} + 8593 p^{12} T^{27} + 911 p^{13} T^{28} + 9 p^{14} T^{29} + p^{15} T^{30} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.52483060603290627409267600389, −2.46996077545237710982589656618, −2.46412121453526570601997138549, −2.23050717859012158112574754915, −2.22453296449998415709268054555, −2.20150822597081405799750352021, −2.17650015995081757498052494437, −2.12457994108672616023204691472, −1.99758401073766028381106387735, −1.89140765381304930866899945832, −1.83246093325833617176675106789, −1.80853634604765283104035343798, −1.67512034700120777553602965382, −1.63793348587671504969627480642, −1.53836183663470679972088717421, −1.53534885353345990287248969977, −1.42767846801256349569037799553, −1.35624209125588488610863109453, −1.30135245085062972735155306577, −1.13090084312646799316111289781, −1.05521339363022844799584140378, −1.03443964799426846646468603822, −1.03154200187763773647899555819, −0.965051357874431407364352787570, −0.816941873324990204489842094945, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.816941873324990204489842094945, 0.965051357874431407364352787570, 1.03154200187763773647899555819, 1.03443964799426846646468603822, 1.05521339363022844799584140378, 1.13090084312646799316111289781, 1.30135245085062972735155306577, 1.35624209125588488610863109453, 1.42767846801256349569037799553, 1.53534885353345990287248969977, 1.53836183663470679972088717421, 1.63793348587671504969627480642, 1.67512034700120777553602965382, 1.80853634604765283104035343798, 1.83246093325833617176675106789, 1.89140765381304930866899945832, 1.99758401073766028381106387735, 2.12457994108672616023204691472, 2.17650015995081757498052494437, 2.20150822597081405799750352021, 2.22453296449998415709268054555, 2.23050717859012158112574754915, 2.46412121453526570601997138549, 2.46996077545237710982589656618, 2.52483060603290627409267600389
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.