Dirichlet series
| L(s) = 1 | − 7·2-s − 4·3-s + 15·4-s − 6·5-s + 28·6-s − 9·7-s + 9·8-s − 14·9-s + 42·10-s − 18·11-s − 60·12-s − 21·13-s + 63·14-s + 24·15-s − 90·16-s − 9·17-s + 98·18-s − 12·19-s − 90·20-s + 36·21-s + 126·22-s + 17·23-s − 36·24-s − 22·25-s + 147·26-s + 81·27-s − 135·28-s + ⋯ |
| L(s) = 1 | − 4.94·2-s − 2.30·3-s + 15/2·4-s − 2.68·5-s + 11.4·6-s − 3.40·7-s + 3.18·8-s − 4.66·9-s + 13.2·10-s − 5.42·11-s − 17.3·12-s − 5.82·13-s + 16.8·14-s + 6.19·15-s − 22.5·16-s − 2.18·17-s + 23.0·18-s − 2.75·19-s − 20.1·20-s + 7.85·21-s + 26.8·22-s + 3.54·23-s − 7.34·24-s − 4.39·25-s + 28.8·26-s + 15.5·27-s − 25.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{17} \cdot 47^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{17} \cdot 47^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(23^{17} \cdot 47^{17}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(8.19874\times 10^{15}\) |
| Root analytic conductor: | \(2.93799\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(17\) |
| Selberg data: | \((34,\ 23^{17} \cdot 47^{17} ,\ ( \ : [1/2]^{17} ),\ -1 )\) |
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 | 23 | \( ( 1 - T )^{17} \) |
| 47 | \( ( 1 - T )^{17} \) | |
| good | 2 | \( 1 + 7 T + 17 p T^{2} + 31 p^{2} T^{3} + 385 T^{4} + 525 p T^{5} + 2609 T^{6} + 749 p^{3} T^{7} + 6447 p T^{8} + 13081 p T^{9} + 50391 T^{10} + 92463 T^{11} + 20289 p^{3} T^{12} + 136535 p T^{13} + 441357 T^{14} + 342913 p T^{15} + 1026047 T^{16} + 1478217 T^{17} + 1026047 p T^{18} + 342913 p^{3} T^{19} + 441357 p^{3} T^{20} + 136535 p^{5} T^{21} + 20289 p^{8} T^{22} + 92463 p^{6} T^{23} + 50391 p^{7} T^{24} + 13081 p^{9} T^{25} + 6447 p^{10} T^{26} + 749 p^{13} T^{27} + 2609 p^{11} T^{28} + 525 p^{13} T^{29} + 385 p^{13} T^{30} + 31 p^{16} T^{31} + 17 p^{16} T^{32} + 7 p^{16} T^{33} + p^{17} T^{34} \) |
| 3 | \( 1 + 4 T + 10 p T^{2} + 95 T^{3} + 140 p T^{4} + 1114 T^{5} + 3724 T^{6} + 8552 T^{7} + 23777 T^{8} + 48385 T^{9} + 117829 T^{10} + 72181 p T^{11} + 479095 T^{12} + 811060 T^{13} + 1682474 T^{14} + 2686423 T^{15} + 5365420 T^{16} + 8269865 T^{17} + 5365420 p T^{18} + 2686423 p^{2} T^{19} + 1682474 p^{3} T^{20} + 811060 p^{4} T^{21} + 479095 p^{5} T^{22} + 72181 p^{7} T^{23} + 117829 p^{7} T^{24} + 48385 p^{8} T^{25} + 23777 p^{9} T^{26} + 8552 p^{10} T^{27} + 3724 p^{11} T^{28} + 1114 p^{12} T^{29} + 140 p^{14} T^{30} + 95 p^{14} T^{31} + 10 p^{16} T^{32} + 4 p^{16} T^{33} + p^{17} T^{34} \) | |
| 5 | \( 1 + 6 T + 58 T^{2} + 272 T^{3} + 1554 T^{4} + 1219 p T^{5} + 5303 p T^{6} + 3627 p^{2} T^{7} + 330021 T^{8} + 40424 p^{2} T^{9} + 3224904 T^{10} + 1799213 p T^{11} + 25851183 T^{12} + 66415846 T^{13} + 174679586 T^{14} + 83208552 p T^{15} + 1011343476 T^{16} + 2239464073 T^{17} + 1011343476 p T^{18} + 83208552 p^{3} T^{19} + 174679586 p^{3} T^{20} + 66415846 p^{4} T^{21} + 25851183 p^{5} T^{22} + 1799213 p^{7} T^{23} + 3224904 p^{7} T^{24} + 40424 p^{10} T^{25} + 330021 p^{9} T^{26} + 3627 p^{12} T^{27} + 5303 p^{12} T^{28} + 1219 p^{13} T^{29} + 1554 p^{13} T^{30} + 272 p^{14} T^{31} + 58 p^{15} T^{32} + 6 p^{16} T^{33} + p^{17} T^{34} \) | |
| 7 | \( 1 + 9 T + 92 T^{2} + 549 T^{3} + 3363 T^{4} + 15303 T^{5} + 70219 T^{6} + 257875 T^{7} + 954606 T^{8} + 2870268 T^{9} + 1250948 p T^{10} + 20839837 T^{11} + 50989702 T^{12} + 79316102 T^{13} + 17730516 p T^{14} - 127797527 T^{15} - 612643585 T^{16} - 3181111263 T^{17} - 612643585 p T^{18} - 127797527 p^{2} T^{19} + 17730516 p^{4} T^{20} + 79316102 p^{4} T^{21} + 50989702 p^{5} T^{22} + 20839837 p^{6} T^{23} + 1250948 p^{8} T^{24} + 2870268 p^{8} T^{25} + 954606 p^{9} T^{26} + 257875 p^{10} T^{27} + 70219 p^{11} T^{28} + 15303 p^{12} T^{29} + 3363 p^{13} T^{30} + 549 p^{14} T^{31} + 92 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 11 | \( 1 + 18 T + 237 T^{2} + 2306 T^{3} + 19059 T^{4} + 134494 T^{5} + 846801 T^{6} + 4771742 T^{7} + 24568937 T^{8} + 115795994 T^{9} + 506130674 T^{10} + 2055768707 T^{11} + 7851587589 T^{12} + 28335564919 T^{13} + 98136074530 T^{14} + 329479458557 T^{15} + 1093335060016 T^{16} + 3615676940341 T^{17} + 1093335060016 p T^{18} + 329479458557 p^{2} T^{19} + 98136074530 p^{3} T^{20} + 28335564919 p^{4} T^{21} + 7851587589 p^{5} T^{22} + 2055768707 p^{6} T^{23} + 506130674 p^{7} T^{24} + 115795994 p^{8} T^{25} + 24568937 p^{9} T^{26} + 4771742 p^{10} T^{27} + 846801 p^{11} T^{28} + 134494 p^{12} T^{29} + 19059 p^{13} T^{30} + 2306 p^{14} T^{31} + 237 p^{15} T^{32} + 18 p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 + 21 T + 332 T^{2} + 3864 T^{3} + 38439 T^{4} + 328383 T^{5} + 2519937 T^{6} + 17457914 T^{7} + 111391866 T^{8} + 656848479 T^{9} + 3618084838 T^{10} + 18654729714 T^{11} + 90609661760 T^{12} + 31924753603 p T^{13} + 1799832407252 T^{14} + 7391232694342 T^{15} + 28816273713568 T^{16} + 106587644533533 T^{17} + 28816273713568 p T^{18} + 7391232694342 p^{2} T^{19} + 1799832407252 p^{3} T^{20} + 31924753603 p^{5} T^{21} + 90609661760 p^{5} T^{22} + 18654729714 p^{6} T^{23} + 3618084838 p^{7} T^{24} + 656848479 p^{8} T^{25} + 111391866 p^{9} T^{26} + 17457914 p^{10} T^{27} + 2519937 p^{11} T^{28} + 328383 p^{12} T^{29} + 38439 p^{13} T^{30} + 3864 p^{14} T^{31} + 332 p^{15} T^{32} + 21 p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 + 9 T + 182 T^{2} + 1110 T^{3} + 13469 T^{4} + 58258 T^{5} + 580860 T^{6} + 1749499 T^{7} + 17804247 T^{8} + 2067362 p T^{9} + 454500968 T^{10} + 555190875 T^{11} + 10574653035 T^{12} + 8054451172 T^{13} + 224751887531 T^{14} + 114897027156 T^{15} + 4276844482415 T^{16} + 1776600565541 T^{17} + 4276844482415 p T^{18} + 114897027156 p^{2} T^{19} + 224751887531 p^{3} T^{20} + 8054451172 p^{4} T^{21} + 10574653035 p^{5} T^{22} + 555190875 p^{6} T^{23} + 454500968 p^{7} T^{24} + 2067362 p^{9} T^{25} + 17804247 p^{9} T^{26} + 1749499 p^{10} T^{27} + 580860 p^{11} T^{28} + 58258 p^{12} T^{29} + 13469 p^{13} T^{30} + 1110 p^{14} T^{31} + 182 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 + 12 T + 11 p T^{2} + 1935 T^{3} + 20784 T^{4} + 161098 T^{5} + 70931 p T^{6} + 9116766 T^{7} + 64558898 T^{8} + 390675447 T^{9} + 2438933470 T^{10} + 13412166155 T^{11} + 75529222415 T^{12} + 381322814192 T^{13} + 1964090589083 T^{14} + 9160350849231 T^{15} + 43509606251642 T^{16} + 187980944795777 T^{17} + 43509606251642 p T^{18} + 9160350849231 p^{2} T^{19} + 1964090589083 p^{3} T^{20} + 381322814192 p^{4} T^{21} + 75529222415 p^{5} T^{22} + 13412166155 p^{6} T^{23} + 2438933470 p^{7} T^{24} + 390675447 p^{8} T^{25} + 64558898 p^{9} T^{26} + 9116766 p^{10} T^{27} + 70931 p^{12} T^{28} + 161098 p^{12} T^{29} + 20784 p^{13} T^{30} + 1935 p^{14} T^{31} + 11 p^{16} T^{32} + 12 p^{16} T^{33} + p^{17} T^{34} \) | |
| 29 | \( 1 + 27 T + 599 T^{2} + 8898 T^{3} + 116384 T^{4} + 1229213 T^{5} + 11916132 T^{6} + 100095368 T^{7} + 798783560 T^{8} + 5763239306 T^{9} + 40711216327 T^{10} + 267108038259 T^{11} + 1746645244922 T^{12} + 10715453014219 T^{13} + 65741508770741 T^{14} + 378443436171507 T^{15} + 2175572678778621 T^{16} + 11726620922542571 T^{17} + 2175572678778621 p T^{18} + 378443436171507 p^{2} T^{19} + 65741508770741 p^{3} T^{20} + 10715453014219 p^{4} T^{21} + 1746645244922 p^{5} T^{22} + 267108038259 p^{6} T^{23} + 40711216327 p^{7} T^{24} + 5763239306 p^{8} T^{25} + 798783560 p^{9} T^{26} + 100095368 p^{10} T^{27} + 11916132 p^{11} T^{28} + 1229213 p^{12} T^{29} + 116384 p^{13} T^{30} + 8898 p^{14} T^{31} + 599 p^{15} T^{32} + 27 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 + 27 T + 593 T^{2} + 8885 T^{3} + 116223 T^{4} + 1244919 T^{5} + 12073312 T^{6} + 102227125 T^{7} + 803772004 T^{8} + 5724648539 T^{9} + 38825258413 T^{10} + 246265433084 T^{11} + 1529921573764 T^{12} + 9143713401638 T^{13} + 54562648816713 T^{14} + 316491040776883 T^{15} + 59041774395308 p T^{16} + 10234049590711707 T^{17} + 59041774395308 p^{2} T^{18} + 316491040776883 p^{2} T^{19} + 54562648816713 p^{3} T^{20} + 9143713401638 p^{4} T^{21} + 1529921573764 p^{5} T^{22} + 246265433084 p^{6} T^{23} + 38825258413 p^{7} T^{24} + 5724648539 p^{8} T^{25} + 803772004 p^{9} T^{26} + 102227125 p^{10} T^{27} + 12073312 p^{11} T^{28} + 1244919 p^{12} T^{29} + 116223 p^{13} T^{30} + 8885 p^{14} T^{31} + 593 p^{15} T^{32} + 27 p^{16} T^{33} + p^{17} T^{34} \) | |
| 37 | \( 1 + 257 T^{2} + 18 T^{3} + 35455 T^{4} - 1423 T^{5} + 3414485 T^{6} - 767229 T^{7} + 256180349 T^{8} - 111801819 T^{9} + 15894837329 T^{10} - 9968880340 T^{11} + 846187521035 T^{12} - 647062227266 T^{13} + 39518006161897 T^{14} - 32785176041838 T^{15} + 1638589139134011 T^{16} - 1342839315107935 T^{17} + 1638589139134011 p T^{18} - 32785176041838 p^{2} T^{19} + 39518006161897 p^{3} T^{20} - 647062227266 p^{4} T^{21} + 846187521035 p^{5} T^{22} - 9968880340 p^{6} T^{23} + 15894837329 p^{7} T^{24} - 111801819 p^{8} T^{25} + 256180349 p^{9} T^{26} - 767229 p^{10} T^{27} + 3414485 p^{11} T^{28} - 1423 p^{12} T^{29} + 35455 p^{13} T^{30} + 18 p^{14} T^{31} + 257 p^{15} T^{32} + p^{17} T^{34} \) | |
| 41 | \( 1 + 44 T + 1297 T^{2} + 27656 T^{3} + 485203 T^{4} + 7172763 T^{5} + 93273133 T^{6} + 1080388427 T^{7} + 11400184348 T^{8} + 110487146997 T^{9} + 998104208763 T^{10} + 8452749202828 T^{11} + 67824134134409 T^{12} + 517486390062986 T^{13} + 3782817752090821 T^{14} + 26529891501495514 T^{15} + 179390430530540061 T^{16} + 1168651706659867057 T^{17} + 179390430530540061 p T^{18} + 26529891501495514 p^{2} T^{19} + 3782817752090821 p^{3} T^{20} + 517486390062986 p^{4} T^{21} + 67824134134409 p^{5} T^{22} + 8452749202828 p^{6} T^{23} + 998104208763 p^{7} T^{24} + 110487146997 p^{8} T^{25} + 11400184348 p^{9} T^{26} + 1080388427 p^{10} T^{27} + 93273133 p^{11} T^{28} + 7172763 p^{12} T^{29} + 485203 p^{13} T^{30} + 27656 p^{14} T^{31} + 1297 p^{15} T^{32} + 44 p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 + 2 T + 256 T^{2} + 177 T^{3} + 32841 T^{4} - 16206 T^{5} + 2968863 T^{6} - 4120665 T^{7} + 219307172 T^{8} - 404937509 T^{9} + 14072064764 T^{10} - 27947359166 T^{11} + 798053645012 T^{12} - 1618819778160 T^{13} + 40495715095928 T^{14} - 83183576796731 T^{15} + 1875365418941183 T^{16} - 3800538523009473 T^{17} + 1875365418941183 p T^{18} - 83183576796731 p^{2} T^{19} + 40495715095928 p^{3} T^{20} - 1618819778160 p^{4} T^{21} + 798053645012 p^{5} T^{22} - 27947359166 p^{6} T^{23} + 14072064764 p^{7} T^{24} - 404937509 p^{8} T^{25} + 219307172 p^{9} T^{26} - 4120665 p^{10} T^{27} + 2968863 p^{11} T^{28} - 16206 p^{12} T^{29} + 32841 p^{13} T^{30} + 177 p^{14} T^{31} + 256 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 - 3 T + 433 T^{2} - 1317 T^{3} + 98311 T^{4} - 314489 T^{5} + 15353981 T^{6} - 52185970 T^{7} + 1837804099 T^{8} - 6606779969 T^{9} + 178438804649 T^{10} - 667956607326 T^{11} + 14539266203223 T^{12} - 55343202973735 T^{13} + 1015586650037464 T^{14} - 3814525040871171 T^{15} + 61603670591268206 T^{16} - 220487535957320329 T^{17} + 61603670591268206 p T^{18} - 3814525040871171 p^{2} T^{19} + 1015586650037464 p^{3} T^{20} - 55343202973735 p^{4} T^{21} + 14539266203223 p^{5} T^{22} - 667956607326 p^{6} T^{23} + 178438804649 p^{7} T^{24} - 6606779969 p^{8} T^{25} + 1837804099 p^{9} T^{26} - 52185970 p^{10} T^{27} + 15353981 p^{11} T^{28} - 314489 p^{12} T^{29} + 98311 p^{13} T^{30} - 1317 p^{14} T^{31} + 433 p^{15} T^{32} - 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 + 24 T + 892 T^{2} + 16220 T^{3} + 347275 T^{4} + 5091417 T^{5} + 80178929 T^{6} + 982470589 T^{7} + 12436036024 T^{8} + 130382638659 T^{9} + 1388610148340 T^{10} + 12700943230051 T^{11} + 117701262139486 T^{12} + 962608670521976 T^{13} + 8069968234593309 T^{14} + 61427217942051009 T^{15} + 490821397183175917 T^{16} + 3652908003904166021 T^{17} + 490821397183175917 p T^{18} + 61427217942051009 p^{2} T^{19} + 8069968234593309 p^{3} T^{20} + 962608670521976 p^{4} T^{21} + 117701262139486 p^{5} T^{22} + 12700943230051 p^{6} T^{23} + 1388610148340 p^{7} T^{24} + 130382638659 p^{8} T^{25} + 12436036024 p^{9} T^{26} + 982470589 p^{10} T^{27} + 80178929 p^{11} T^{28} + 5091417 p^{12} T^{29} + 347275 p^{13} T^{30} + 16220 p^{14} T^{31} + 892 p^{15} T^{32} + 24 p^{16} T^{33} + p^{17} T^{34} \) | |
| 61 | \( 1 + 3 T + 467 T^{2} + 1641 T^{3} + 116031 T^{4} + 474977 T^{5} + 20213014 T^{6} + 91600409 T^{7} + 2740991775 T^{8} + 13170775342 T^{9} + 304946072485 T^{10} + 1501082734929 T^{11} + 28646940983251 T^{12} + 140387378564482 T^{13} + 2311808508420017 T^{14} + 10991072135123787 T^{15} + 161840342591203212 T^{16} + 727798004207801163 T^{17} + 161840342591203212 p T^{18} + 10991072135123787 p^{2} T^{19} + 2311808508420017 p^{3} T^{20} + 140387378564482 p^{4} T^{21} + 28646940983251 p^{5} T^{22} + 1501082734929 p^{6} T^{23} + 304946072485 p^{7} T^{24} + 13170775342 p^{8} T^{25} + 2740991775 p^{9} T^{26} + 91600409 p^{10} T^{27} + 20213014 p^{11} T^{28} + 474977 p^{12} T^{29} + 116031 p^{13} T^{30} + 1641 p^{14} T^{31} + 467 p^{15} T^{32} + 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 - 11 T + 496 T^{2} - 6630 T^{3} + 145535 T^{4} - 1946330 T^{5} + 31253011 T^{6} - 389285729 T^{7} + 5199150020 T^{8} - 59602260406 T^{9} + 695910720236 T^{10} - 7345220087019 T^{11} + 76963356879200 T^{12} - 748829965476819 T^{13} + 7146282741007756 T^{14} - 64189292915257394 T^{15} + 562355369227417744 T^{16} - 4664937178600344723 T^{17} + 562355369227417744 p T^{18} - 64189292915257394 p^{2} T^{19} + 7146282741007756 p^{3} T^{20} - 748829965476819 p^{4} T^{21} + 76963356879200 p^{5} T^{22} - 7345220087019 p^{6} T^{23} + 695910720236 p^{7} T^{24} - 59602260406 p^{8} T^{25} + 5199150020 p^{9} T^{26} - 389285729 p^{10} T^{27} + 31253011 p^{11} T^{28} - 1946330 p^{12} T^{29} + 145535 p^{13} T^{30} - 6630 p^{14} T^{31} + 496 p^{15} T^{32} - 11 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 + 87 T + 4309 T^{2} + 153572 T^{3} + 4358839 T^{4} + 103809399 T^{5} + 2144245335 T^{6} + 39265470196 T^{7} + 647597106700 T^{8} + 9731899971397 T^{9} + 134444562010063 T^{10} + 1719125075974500 T^{11} + 20455425741577204 T^{12} + 227426926975958393 T^{13} + 2370205154791887236 T^{14} + 23209040403497440361 T^{15} + \)\(21\!\cdots\!88\)\( T^{16} + \)\(18\!\cdots\!91\)\( T^{17} + \)\(21\!\cdots\!88\)\( p T^{18} + 23209040403497440361 p^{2} T^{19} + 2370205154791887236 p^{3} T^{20} + 227426926975958393 p^{4} T^{21} + 20455425741577204 p^{5} T^{22} + 1719125075974500 p^{6} T^{23} + 134444562010063 p^{7} T^{24} + 9731899971397 p^{8} T^{25} + 647597106700 p^{9} T^{26} + 39265470196 p^{10} T^{27} + 2144245335 p^{11} T^{28} + 103809399 p^{12} T^{29} + 4358839 p^{13} T^{30} + 153572 p^{14} T^{31} + 4309 p^{15} T^{32} + 87 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 + 48 T + 1857 T^{2} + 48911 T^{3} + 1121282 T^{4} + 20976959 T^{5} + 355288334 T^{6} + 5239302580 T^{7} + 71914527200 T^{8} + 891464519392 T^{9} + 10535311376494 T^{10} + 115397362181527 T^{11} + 1228635375968432 T^{12} + 12320621574362027 T^{13} + 121245072814098176 T^{14} + 1128863821862870342 T^{15} + 10314369490950366936 T^{16} + 89029054215721108108 T^{17} + 10314369490950366936 p T^{18} + 1128863821862870342 p^{2} T^{19} + 121245072814098176 p^{3} T^{20} + 12320621574362027 p^{4} T^{21} + 1228635375968432 p^{5} T^{22} + 115397362181527 p^{6} T^{23} + 10535311376494 p^{7} T^{24} + 891464519392 p^{8} T^{25} + 71914527200 p^{9} T^{26} + 5239302580 p^{10} T^{27} + 355288334 p^{11} T^{28} + 20976959 p^{12} T^{29} + 1121282 p^{13} T^{30} + 48911 p^{14} T^{31} + 1857 p^{15} T^{32} + 48 p^{16} T^{33} + p^{17} T^{34} \) | |
| 79 | \( 1 + 53 T + 1915 T^{2} + 49329 T^{3} + 1046518 T^{4} + 18477503 T^{5} + 285033293 T^{6} + 3842963580 T^{7} + 46263452722 T^{8} + 492320457149 T^{9} + 4638063942432 T^{10} + 37174395506472 T^{11} + 236168968712481 T^{12} + 785395003077062 T^{13} - 6567689336507239 T^{14} - 174169323315131026 T^{15} - 2270359094646299096 T^{16} - 22615816087002032905 T^{17} - 2270359094646299096 p T^{18} - 174169323315131026 p^{2} T^{19} - 6567689336507239 p^{3} T^{20} + 785395003077062 p^{4} T^{21} + 236168968712481 p^{5} T^{22} + 37174395506472 p^{6} T^{23} + 4638063942432 p^{7} T^{24} + 492320457149 p^{8} T^{25} + 46263452722 p^{9} T^{26} + 3842963580 p^{10} T^{27} + 285033293 p^{11} T^{28} + 18477503 p^{12} T^{29} + 1046518 p^{13} T^{30} + 49329 p^{14} T^{31} + 1915 p^{15} T^{32} + 53 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 + 22 T + 992 T^{2} + 18517 T^{3} + 480864 T^{4} + 7802167 T^{5} + 151796922 T^{6} + 2180995211 T^{7} + 35022616103 T^{8} + 451791137103 T^{9} + 6276814112638 T^{10} + 73408881849564 T^{11} + 905809618416396 T^{12} + 9665648781159925 T^{13} + 107580433436903764 T^{14} + 1051110868401156056 T^{15} + 10649867600978250791 T^{16} + 95363308212425509155 T^{17} + 10649867600978250791 p T^{18} + 1051110868401156056 p^{2} T^{19} + 107580433436903764 p^{3} T^{20} + 9665648781159925 p^{4} T^{21} + 905809618416396 p^{5} T^{22} + 73408881849564 p^{6} T^{23} + 6276814112638 p^{7} T^{24} + 451791137103 p^{8} T^{25} + 35022616103 p^{9} T^{26} + 2180995211 p^{10} T^{27} + 151796922 p^{11} T^{28} + 7802167 p^{12} T^{29} + 480864 p^{13} T^{30} + 18517 p^{14} T^{31} + 992 p^{15} T^{32} + 22 p^{16} T^{33} + p^{17} T^{34} \) | |
| 89 | \( 1 - 9 T + 757 T^{2} - 7961 T^{3} + 298263 T^{4} - 3353878 T^{5} + 81128073 T^{6} - 918161831 T^{7} + 16910768295 T^{8} - 185906600301 T^{9} + 2843511249534 T^{10} - 29831764131518 T^{11} + 397895268281265 T^{12} - 3950262903516380 T^{13} + 47331282025260652 T^{14} - 442249437164655824 T^{15} + 4853252712656012898 T^{16} - 42426810684914666355 T^{17} + 4853252712656012898 p T^{18} - 442249437164655824 p^{2} T^{19} + 47331282025260652 p^{3} T^{20} - 3950262903516380 p^{4} T^{21} + 397895268281265 p^{5} T^{22} - 29831764131518 p^{6} T^{23} + 2843511249534 p^{7} T^{24} - 185906600301 p^{8} T^{25} + 16910768295 p^{9} T^{26} - 918161831 p^{10} T^{27} + 81128073 p^{11} T^{28} - 3353878 p^{12} T^{29} + 298263 p^{13} T^{30} - 7961 p^{14} T^{31} + 757 p^{15} T^{32} - 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 97 | \( 1 + 6 T + 754 T^{2} + 3833 T^{3} + 277285 T^{4} + 1218220 T^{5} + 67723039 T^{6} + 267313675 T^{7} + 12579743161 T^{8} + 47192635409 T^{9} + 1920413573643 T^{10} + 7250474398315 T^{11} + 253078200144393 T^{12} + 991245193018101 T^{13} + 29695523238830054 T^{14} + 119324061063738108 T^{15} + 3156362135050624937 T^{16} + 12461271176562443769 T^{17} + 3156362135050624937 p T^{18} + 119324061063738108 p^{2} T^{19} + 29695523238830054 p^{3} T^{20} + 991245193018101 p^{4} T^{21} + 253078200144393 p^{5} T^{22} + 7250474398315 p^{6} T^{23} + 1920413573643 p^{7} T^{24} + 47192635409 p^{8} T^{25} + 12579743161 p^{9} T^{26} + 267313675 p^{10} T^{27} + 67723039 p^{11} T^{28} + 1218220 p^{12} T^{29} + 277285 p^{13} T^{30} + 3833 p^{14} T^{31} + 754 p^{15} T^{32} + 6 p^{16} T^{33} + p^{17} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.09160033708806196014984773229, −2.95351520461142957188905362845, −2.80232281486999710507811840184, −2.71559613375350978212347971493, −2.71062001988938378193899020247, −2.66189877238415573366604397441, −2.65970372611014528505108522234, −2.57779916720157167593666882495, −2.51715501558678375130074860487, −2.27697624340277434816931431075, −2.25573527741423241733534755298, −2.21603334361577884260428865260, −2.19649844677681356171312241379, −2.19461949869266826404336670409, −2.12894037233996114591009552917, −1.84146939737491879591841202491, −1.83885182849558796549498257851, −1.60151424489967532446912568465, −1.58744314598833147828747896397, −1.55280246303699536177807586112, −1.51362790140707443522489843114, −1.33738980187932854182933353400, −1.22953486405852282452761871512, −1.12375946633690171677285037703, −0.850908494636670131091128791476, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.850908494636670131091128791476, 1.12375946633690171677285037703, 1.22953486405852282452761871512, 1.33738980187932854182933353400, 1.51362790140707443522489843114, 1.55280246303699536177807586112, 1.58744314598833147828747896397, 1.60151424489967532446912568465, 1.83885182849558796549498257851, 1.84146939737491879591841202491, 2.12894037233996114591009552917, 2.19461949869266826404336670409, 2.19649844677681356171312241379, 2.21603334361577884260428865260, 2.25573527741423241733534755298, 2.27697624340277434816931431075, 2.51715501558678375130074860487, 2.57779916720157167593666882495, 2.65970372611014528505108522234, 2.66189877238415573366604397441, 2.71062001988938378193899020247, 2.71559613375350978212347971493, 2.80232281486999710507811840184, 2.95351520461142957188905362845, 3.09160033708806196014984773229
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.