Dirichlet series
| L(s) = 1 | + 15·2-s − 9·3-s + 120·4-s − 11·5-s − 135·6-s − 11·7-s + 680·8-s + 23·9-s − 165·10-s + 15·11-s − 1.08e3·12-s − 21·13-s − 165·14-s + 99·15-s + 3.06e3·16-s − 4·17-s + 345·18-s − 22·19-s − 1.32e3·20-s + 99·21-s + 225·22-s − 16·23-s − 6.12e3·24-s + 26·25-s − 315·26-s + 38·27-s − 1.32e3·28-s + ⋯ |
| L(s) = 1 | + 10.6·2-s − 5.19·3-s + 60·4-s − 4.91·5-s − 55.1·6-s − 4.15·7-s + 240.·8-s + 23/3·9-s − 52.1·10-s + 4.52·11-s − 311.·12-s − 5.82·13-s − 44.0·14-s + 25.5·15-s + 765·16-s − 0.970·17-s + 81.3·18-s − 5.04·19-s − 295.·20-s + 21.6·21-s + 47.9·22-s − 3.33·23-s − 1.24e3·24-s + 26/5·25-s − 61.7·26-s + 7.31·27-s − 249.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 11^{15} \cdot 197^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 11^{15} \cdot 197^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{15} \cdot 11^{15} \cdot 197^{15}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.22317\times 10^{23}\) |
| Root analytic conductor: | \(5.88278\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(15\) |
| Selberg data: | \((30,\ 2^{15} \cdot 11^{15} \cdot 197^{15} ,\ ( \ : [1/2]^{15} ),\ -1 )\) |
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 | 2 | \( ( 1 - T )^{15} \) |
| 11 | \( ( 1 - T )^{15} \) | |
| 197 | \( ( 1 + T )^{15} \) | |
| good | 3 | \( 1 + p^{2} T + 58 T^{2} + 277 T^{3} + 1124 T^{4} + 3923 T^{5} + 4100 p T^{6} + 34883 T^{7} + 91351 T^{8} + 221872 T^{9} + 505961 T^{10} + 1086365 T^{11} + 246043 p^{2} T^{12} + 4290925 T^{13} + 7947562 T^{14} + 14063192 T^{15} + 7947562 p T^{16} + 4290925 p^{2} T^{17} + 246043 p^{5} T^{18} + 1086365 p^{4} T^{19} + 505961 p^{5} T^{20} + 221872 p^{6} T^{21} + 91351 p^{7} T^{22} + 34883 p^{8} T^{23} + 4100 p^{10} T^{24} + 3923 p^{10} T^{25} + 1124 p^{11} T^{26} + 277 p^{12} T^{27} + 58 p^{13} T^{28} + p^{16} T^{29} + p^{15} T^{30} \) |
| 5 | \( 1 + 11 T + 19 p T^{2} + 24 p^{2} T^{3} + 3287 T^{4} + 15419 T^{5} + 65406 T^{6} + 249972 T^{7} + 176677 p T^{8} + 2881542 T^{9} + 8809022 T^{10} + 25198016 T^{11} + 68126179 T^{12} + 34742468 p T^{13} + 84138511 p T^{14} + 964479007 T^{15} + 84138511 p^{2} T^{16} + 34742468 p^{3} T^{17} + 68126179 p^{3} T^{18} + 25198016 p^{4} T^{19} + 8809022 p^{5} T^{20} + 2881542 p^{6} T^{21} + 176677 p^{8} T^{22} + 249972 p^{8} T^{23} + 65406 p^{9} T^{24} + 15419 p^{10} T^{25} + 3287 p^{11} T^{26} + 24 p^{14} T^{27} + 19 p^{14} T^{28} + 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 + 11 T + 16 p T^{2} + 748 T^{3} + 4636 T^{4} + 23382 T^{5} + 111465 T^{6} + 467373 T^{7} + 1875005 T^{8} + 6861332 T^{9} + 24156859 T^{10} + 79006009 T^{11} + 35597304 p T^{12} + 737150515 T^{13} + 2105820574 T^{14} + 5668149663 T^{15} + 2105820574 p T^{16} + 737150515 p^{2} T^{17} + 35597304 p^{4} T^{18} + 79006009 p^{4} T^{19} + 24156859 p^{5} T^{20} + 6861332 p^{6} T^{21} + 1875005 p^{7} T^{22} + 467373 p^{8} T^{23} + 111465 p^{9} T^{24} + 23382 p^{10} T^{25} + 4636 p^{11} T^{26} + 748 p^{12} T^{27} + 16 p^{14} T^{28} + 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 + 21 T + 318 T^{2} + 3451 T^{3} + 31583 T^{4} + 243390 T^{5} + 1671777 T^{6} + 10223672 T^{7} + 57356293 T^{8} + 294834327 T^{9} + 1416834648 T^{10} + 488960993 p T^{11} + 27042829106 T^{12} + 108859746713 T^{13} + 419773391816 T^{14} + 1543417653594 T^{15} + 419773391816 p T^{16} + 108859746713 p^{2} T^{17} + 27042829106 p^{3} T^{18} + 488960993 p^{5} T^{19} + 1416834648 p^{5} T^{20} + 294834327 p^{6} T^{21} + 57356293 p^{7} T^{22} + 10223672 p^{8} T^{23} + 1671777 p^{9} T^{24} + 243390 p^{10} T^{25} + 31583 p^{11} T^{26} + 3451 p^{12} T^{27} + 318 p^{13} T^{28} + 21 p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 + 4 T + 137 T^{2} + 339 T^{3} + 514 p T^{4} + 11973 T^{5} + 371031 T^{6} + 211846 T^{7} + 12216747 T^{8} + 312586 T^{9} + 331738801 T^{10} - 105120332 T^{11} + 7619064070 T^{12} - 3731536184 T^{13} + 150098427900 T^{14} - 77504253095 T^{15} + 150098427900 p T^{16} - 3731536184 p^{2} T^{17} + 7619064070 p^{3} T^{18} - 105120332 p^{4} T^{19} + 331738801 p^{5} T^{20} + 312586 p^{6} T^{21} + 12216747 p^{7} T^{22} + 211846 p^{8} T^{23} + 371031 p^{9} T^{24} + 11973 p^{10} T^{25} + 514 p^{12} T^{26} + 339 p^{12} T^{27} + 137 p^{13} T^{28} + 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 + 22 T + 20 p T^{2} + 4585 T^{3} + 47944 T^{4} + 415099 T^{5} + 3237706 T^{6} + 22096599 T^{7} + 138783310 T^{8} + 783389853 T^{9} + 4142522345 T^{10} + 20107372457 T^{11} + 93771413152 T^{12} + 413406390269 T^{13} + 1820458003058 T^{14} + 7842761383788 T^{15} + 1820458003058 p T^{16} + 413406390269 p^{2} T^{17} + 93771413152 p^{3} T^{18} + 20107372457 p^{4} T^{19} + 4142522345 p^{5} T^{20} + 783389853 p^{6} T^{21} + 138783310 p^{7} T^{22} + 22096599 p^{8} T^{23} + 3237706 p^{9} T^{24} + 415099 p^{10} T^{25} + 47944 p^{11} T^{26} + 4585 p^{12} T^{27} + 20 p^{14} T^{28} + 22 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 + 16 T + 256 T^{2} + 2842 T^{3} + 29393 T^{4} + 261767 T^{5} + 2164633 T^{6} + 16390426 T^{7} + 116403509 T^{8} + 773990961 T^{9} + 4873021647 T^{10} + 29003052564 T^{11} + 164481834039 T^{12} + 886140158281 T^{13} + 4563241377492 T^{14} + 22373714393314 T^{15} + 4563241377492 p T^{16} + 886140158281 p^{2} T^{17} + 164481834039 p^{3} T^{18} + 29003052564 p^{4} T^{19} + 4873021647 p^{5} T^{20} + 773990961 p^{6} T^{21} + 116403509 p^{7} T^{22} + 16390426 p^{8} T^{23} + 2164633 p^{9} T^{24} + 261767 p^{10} T^{25} + 29393 p^{11} T^{26} + 2842 p^{12} T^{27} + 256 p^{13} T^{28} + 16 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 + 8 T + 295 T^{2} + 2247 T^{3} + 44294 T^{4} + 313607 T^{5} + 4392088 T^{6} + 28662652 T^{7} + 318444693 T^{8} + 1907034711 T^{9} + 17783476462 T^{10} + 97387651745 T^{11} + 787675241969 T^{12} + 3928007878361 T^{13} + 28126095415877 T^{14} + 126963741744927 T^{15} + 28126095415877 p T^{16} + 3928007878361 p^{2} T^{17} + 787675241969 p^{3} T^{18} + 97387651745 p^{4} T^{19} + 17783476462 p^{5} T^{20} + 1907034711 p^{6} T^{21} + 318444693 p^{7} T^{22} + 28662652 p^{8} T^{23} + 4392088 p^{9} T^{24} + 313607 p^{10} T^{25} + 44294 p^{11} T^{26} + 2247 p^{12} T^{27} + 295 p^{13} T^{28} + 8 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + 33 T + 737 T^{2} + 12284 T^{3} + 171923 T^{4} + 2077487 T^{5} + 22453600 T^{6} + 219697872 T^{7} + 1976557638 T^{8} + 16450994119 T^{9} + 127710201385 T^{10} + 927685332832 T^{11} + 6334848016061 T^{12} + 40724149995221 T^{13} + 247099128098580 T^{14} + 1415247047474245 T^{15} + 247099128098580 p T^{16} + 40724149995221 p^{2} T^{17} + 6334848016061 p^{3} T^{18} + 927685332832 p^{4} T^{19} + 127710201385 p^{5} T^{20} + 16450994119 p^{6} T^{21} + 1976557638 p^{7} T^{22} + 219697872 p^{8} T^{23} + 22453600 p^{9} T^{24} + 2077487 p^{10} T^{25} + 171923 p^{11} T^{26} + 12284 p^{12} T^{27} + 737 p^{13} T^{28} + 33 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 + T + 309 T^{2} - 11 T^{3} + 44832 T^{4} - 53498 T^{5} + 4090160 T^{6} - 10547602 T^{7} + 265961001 T^{8} - 1143407143 T^{9} + 360765981 p T^{10} - 84925462371 T^{11} + 556865952393 T^{12} - 4675640781003 T^{13} + 21068760875439 T^{14} - 196943036747566 T^{15} + 21068760875439 p T^{16} - 4675640781003 p^{2} T^{17} + 556865952393 p^{3} T^{18} - 84925462371 p^{4} T^{19} + 360765981 p^{6} T^{20} - 1143407143 p^{6} T^{21} + 265961001 p^{7} T^{22} - 10547602 p^{8} T^{23} + 4090160 p^{9} T^{24} - 53498 p^{10} T^{25} + 44832 p^{11} T^{26} - 11 p^{12} T^{27} + 309 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 10 T + 383 T^{2} + 3738 T^{3} + 73310 T^{4} + 681674 T^{5} + 9264822 T^{6} + 81050269 T^{7} + 863716432 T^{8} + 7062896875 T^{9} + 63020333850 T^{10} + 479624226826 T^{11} + 3728929819069 T^{12} + 26287034648004 T^{13} + 182745469517475 T^{14} + 1184883185802280 T^{15} + 182745469517475 p T^{16} + 26287034648004 p^{2} T^{17} + 3728929819069 p^{3} T^{18} + 479624226826 p^{4} T^{19} + 63020333850 p^{5} T^{20} + 7062896875 p^{6} T^{21} + 863716432 p^{7} T^{22} + 81050269 p^{8} T^{23} + 9264822 p^{9} T^{24} + 681674 p^{10} T^{25} + 73310 p^{11} T^{26} + 3738 p^{12} T^{27} + 383 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 + 8 T + 397 T^{2} + 2987 T^{3} + 75326 T^{4} + 536141 T^{5} + 9038598 T^{6} + 61584050 T^{7} + 769976744 T^{8} + 5106675941 T^{9} + 50064076643 T^{10} + 329479247176 T^{11} + 2654119095264 T^{12} + 17537687445663 T^{13} + 123219977634353 T^{14} + 804992515301388 T^{15} + 123219977634353 p T^{16} + 17537687445663 p^{2} T^{17} + 2654119095264 p^{3} T^{18} + 329479247176 p^{4} T^{19} + 50064076643 p^{5} T^{20} + 5106675941 p^{6} T^{21} + 769976744 p^{7} T^{22} + 61584050 p^{8} T^{23} + 9038598 p^{9} T^{24} + 536141 p^{10} T^{25} + 75326 p^{11} T^{26} + 2987 p^{12} T^{27} + 397 p^{13} T^{28} + 8 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + 31 T + 15 p T^{2} + 11381 T^{3} + 156883 T^{4} + 1802741 T^{5} + 18456621 T^{6} + 164697265 T^{7} + 1326515289 T^{8} + 9373081194 T^{9} + 1257195553 p T^{10} + 313685318899 T^{11} + 1367205221204 T^{12} + 3685130331387 T^{13} - 1397633385038 T^{14} - 87367235841768 T^{15} - 1397633385038 p T^{16} + 3685130331387 p^{2} T^{17} + 1367205221204 p^{3} T^{18} + 313685318899 p^{4} T^{19} + 1257195553 p^{6} T^{20} + 9373081194 p^{6} T^{21} + 1326515289 p^{7} T^{22} + 164697265 p^{8} T^{23} + 18456621 p^{9} T^{24} + 1802741 p^{10} T^{25} + 156883 p^{11} T^{26} + 11381 p^{12} T^{27} + 15 p^{14} T^{28} + 31 p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 + 18 T + 491 T^{2} + 7625 T^{3} + 130087 T^{4} + 1705066 T^{5} + 22878948 T^{6} + 260284369 T^{7} + 2953626349 T^{8} + 29732008385 T^{9} + 295374594376 T^{10} + 2662156115576 T^{11} + 23577366319323 T^{12} + 191643018764100 T^{13} + 1527689576064443 T^{14} + 11233622608828106 T^{15} + 1527689576064443 p T^{16} + 191643018764100 p^{2} T^{17} + 23577366319323 p^{3} T^{18} + 2662156115576 p^{4} T^{19} + 295374594376 p^{5} T^{20} + 29732008385 p^{6} T^{21} + 2953626349 p^{7} T^{22} + 260284369 p^{8} T^{23} + 22878948 p^{9} T^{24} + 1705066 p^{10} T^{25} + 130087 p^{11} T^{26} + 7625 p^{12} T^{27} + 491 p^{13} T^{28} + 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 + 37 T + 1133 T^{2} + 23995 T^{3} + 446042 T^{4} + 6888719 T^{5} + 96857335 T^{6} + 1206189444 T^{7} + 14006419501 T^{8} + 148794068144 T^{9} + 1498272598772 T^{10} + 14068911336482 T^{11} + 126673397795805 T^{12} + 1076535151330845 T^{13} + 8841011289720021 T^{14} + 1169149362079825 p T^{15} + 8841011289720021 p T^{16} + 1076535151330845 p^{2} T^{17} + 126673397795805 p^{3} T^{18} + 14068911336482 p^{4} T^{19} + 1498272598772 p^{5} T^{20} + 148794068144 p^{6} T^{21} + 14006419501 p^{7} T^{22} + 1206189444 p^{8} T^{23} + 96857335 p^{9} T^{24} + 6888719 p^{10} T^{25} + 446042 p^{11} T^{26} + 23995 p^{12} T^{27} + 1133 p^{13} T^{28} + 37 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 31 T + 1116 T^{2} + 23528 T^{3} + 506875 T^{4} + 8286466 T^{5} + 134331471 T^{6} + 1812127496 T^{7} + 393850239 p T^{8} + 276904234420 T^{9} + 3128692003022 T^{10} + 31459771790665 T^{11} + 309986375818265 T^{12} + 2752823259929967 T^{13} + 23959948449377387 T^{14} + 189011903566185991 T^{15} + 23959948449377387 p T^{16} + 2752823259929967 p^{2} T^{17} + 309986375818265 p^{3} T^{18} + 31459771790665 p^{4} T^{19} + 3128692003022 p^{5} T^{20} + 276904234420 p^{6} T^{21} + 393850239 p^{8} T^{22} + 1812127496 p^{8} T^{23} + 134331471 p^{9} T^{24} + 8286466 p^{10} T^{25} + 506875 p^{11} T^{26} + 23528 p^{12} T^{27} + 1116 p^{13} T^{28} + 31 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 - T + 697 T^{2} - 1217 T^{3} + 226052 T^{4} - 616312 T^{5} + 45344278 T^{6} - 180759847 T^{7} + 6334123930 T^{8} - 35135265011 T^{9} + 664004781115 T^{10} - 4853525465064 T^{11} + 55937831711795 T^{12} - 495333789370377 T^{13} + 4080215954754752 T^{14} - 38105666797800458 T^{15} + 4080215954754752 p T^{16} - 495333789370377 p^{2} T^{17} + 55937831711795 p^{3} T^{18} - 4853525465064 p^{4} T^{19} + 664004781115 p^{5} T^{20} - 35135265011 p^{6} T^{21} + 6334123930 p^{7} T^{22} - 180759847 p^{8} T^{23} + 45344278 p^{9} T^{24} - 616312 p^{10} T^{25} + 226052 p^{11} T^{26} - 1217 p^{12} T^{27} + 697 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 + 28 T + 888 T^{2} + 17402 T^{3} + 345198 T^{4} + 5403192 T^{5} + 83517915 T^{6} + 1110154473 T^{7} + 14465290199 T^{8} + 168663071019 T^{9} + 1923170508250 T^{10} + 20033231892431 T^{11} + 203925995882138 T^{12} + 1917162927500295 T^{13} + 17610851954717370 T^{14} + 150121307666332733 T^{15} + 17610851954717370 p T^{16} + 1917162927500295 p^{2} T^{17} + 203925995882138 p^{3} T^{18} + 20033231892431 p^{4} T^{19} + 1923170508250 p^{5} T^{20} + 168663071019 p^{6} T^{21} + 14465290199 p^{7} T^{22} + 1110154473 p^{8} T^{23} + 83517915 p^{9} T^{24} + 5403192 p^{10} T^{25} + 345198 p^{11} T^{26} + 17402 p^{12} T^{27} + 888 p^{13} T^{28} + 28 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 20 T + 792 T^{2} + 12046 T^{3} + 279902 T^{4} + 3516304 T^{5} + 61664621 T^{6} + 668287575 T^{7} + 9732241849 T^{8} + 93501255983 T^{9} + 1189983880056 T^{10} + 10340722258171 T^{11} + 118796864430878 T^{12} + 948650570251887 T^{13} + 10041918721181842 T^{14} + 74458716685439835 T^{15} + 10041918721181842 p T^{16} + 948650570251887 p^{2} T^{17} + 118796864430878 p^{3} T^{18} + 10340722258171 p^{4} T^{19} + 1189983880056 p^{5} T^{20} + 93501255983 p^{6} T^{21} + 9732241849 p^{7} T^{22} + 668287575 p^{8} T^{23} + 61664621 p^{9} T^{24} + 3516304 p^{10} T^{25} + 279902 p^{11} T^{26} + 12046 p^{12} T^{27} + 792 p^{13} T^{28} + 20 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 + 6 T + 617 T^{2} + 4866 T^{3} + 195908 T^{4} + 1764612 T^{5} + 43166035 T^{6} + 402695097 T^{7} + 7326701180 T^{8} + 66904782630 T^{9} + 1000893533951 T^{10} + 8691618038099 T^{11} + 112722345450370 T^{12} + 914470021642513 T^{13} + 10606141402147674 T^{14} + 79325165063665094 T^{15} + 10606141402147674 p T^{16} + 914470021642513 p^{2} T^{17} + 112722345450370 p^{3} T^{18} + 8691618038099 p^{4} T^{19} + 1000893533951 p^{5} T^{20} + 66904782630 p^{6} T^{21} + 7326701180 p^{7} T^{22} + 402695097 p^{8} T^{23} + 43166035 p^{9} T^{24} + 1764612 p^{10} T^{25} + 195908 p^{11} T^{26} + 4866 p^{12} T^{27} + 617 p^{13} T^{28} + 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 + 15 T + 770 T^{2} + 10986 T^{3} + 296544 T^{4} + 3926173 T^{5} + 75650877 T^{6} + 918485225 T^{7} + 14292545189 T^{8} + 158654225504 T^{9} + 2119597955270 T^{10} + 21539491861211 T^{11} + 255113809393402 T^{12} + 2376493925240115 T^{13} + 25390926895099129 T^{14} + 216541456289141522 T^{15} + 25390926895099129 p T^{16} + 2376493925240115 p^{2} T^{17} + 255113809393402 p^{3} T^{18} + 21539491861211 p^{4} T^{19} + 2119597955270 p^{5} T^{20} + 158654225504 p^{6} T^{21} + 14292545189 p^{7} T^{22} + 918485225 p^{8} T^{23} + 75650877 p^{9} T^{24} + 3926173 p^{10} T^{25} + 296544 p^{11} T^{26} + 10986 p^{12} T^{27} + 770 p^{13} T^{28} + 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 17 T + 611 T^{2} + 8434 T^{3} + 180816 T^{4} + 2202847 T^{5} + 36739639 T^{6} + 406811541 T^{7} + 5799046358 T^{8} + 59431379419 T^{9} + 759858437497 T^{10} + 7319167519336 T^{11} + 86171115818344 T^{12} + 783807499647762 T^{13} + 8622298585457082 T^{14} + 74053558023796412 T^{15} + 8622298585457082 p T^{16} + 783807499647762 p^{2} T^{17} + 86171115818344 p^{3} T^{18} + 7319167519336 p^{4} T^{19} + 759858437497 p^{5} T^{20} + 59431379419 p^{6} T^{21} + 5799046358 p^{7} T^{22} + 406811541 p^{8} T^{23} + 36739639 p^{9} T^{24} + 2202847 p^{10} T^{25} + 180816 p^{11} T^{26} + 8434 p^{12} T^{27} + 611 p^{13} T^{28} + 17 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 + 9 T + 707 T^{2} + 4654 T^{3} + 244499 T^{4} + 1278395 T^{5} + 57943986 T^{6} + 258034852 T^{7} + 10635969175 T^{8} + 42174831542 T^{9} + 1601711294489 T^{10} + 5836253042763 T^{11} + 204879174011194 T^{12} + 696184518807301 T^{13} + 22694078461001398 T^{14} + 72183880245396845 T^{15} + 22694078461001398 p T^{16} + 696184518807301 p^{2} T^{17} + 204879174011194 p^{3} T^{18} + 5836253042763 p^{4} T^{19} + 1601711294489 p^{5} T^{20} + 42174831542 p^{6} T^{21} + 10635969175 p^{7} T^{22} + 258034852 p^{8} T^{23} + 57943986 p^{9} T^{24} + 1278395 p^{10} T^{25} + 244499 p^{11} T^{26} + 4654 p^{12} T^{27} + 707 p^{13} T^{28} + 9 p^{14} T^{29} + p^{15} T^{30} \) | |
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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
−2.64031655348553459827962864968, −2.61734810870303116564733118178, −2.56310785487674215424947584918, −2.53394318467943806359038929133, −2.47256297441193377488983948986, −2.43394900406766749581138496561, −2.41720553406537893157437068713, −2.41660358660051405878645270511, −2.37559831065828186784863551237, −2.12419093739434142491930903406, −1.94932008746797559190474025871, −1.92074364336602407451782480465, −1.87429862561475622461409181621, −1.81079894920027954095094012425, −1.80800894149388536422592047430, −1.71545191568509784810292188667, −1.59295623314492827155352958095, −1.55773691870504686819821925739, −1.54950453361540909120575379486, −1.35858845364839394079704419901, −1.31616983361749065643852744127, −1.29140151885927141111934702938, −1.27269661811851318428856278514, −1.22546583053122462712019369046, −0.965791379265017190297019750906, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.965791379265017190297019750906, 1.22546583053122462712019369046, 1.27269661811851318428856278514, 1.29140151885927141111934702938, 1.31616983361749065643852744127, 1.35858845364839394079704419901, 1.54950453361540909120575379486, 1.55773691870504686819821925739, 1.59295623314492827155352958095, 1.71545191568509784810292188667, 1.80800894149388536422592047430, 1.81079894920027954095094012425, 1.87429862561475622461409181621, 1.92074364336602407451782480465, 1.94932008746797559190474025871, 2.12419093739434142491930903406, 2.37559831065828186784863551237, 2.41660358660051405878645270511, 2.41720553406537893157437068713, 2.43394900406766749581138496561, 2.47256297441193377488983948986, 2.53394318467943806359038929133, 2.56310785487674215424947584918, 2.61734810870303116564733118178, 2.64031655348553459827962864968
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.