Dirichlet series
| L(s) = 1 | − 89·3-s − 1.02e3·4-s − 2.38e3·5-s − 9.89e3·7-s + 1.16e4·8-s − 4.83e4·9-s − 7.84e4·11-s + 9.11e4·12-s − 2.96e5·13-s + 2.12e5·15-s + 2.64e5·16-s − 1.12e6·17-s − 1.30e6·19-s + 2.44e6·20-s + 8.80e5·21-s − 1.95e6·23-s − 1.03e6·24-s − 4.64e6·25-s + 6.11e6·27-s + 1.01e7·28-s + 2.81e6·29-s + 7.33e6·31-s − 1.02e7·32-s + 6.98e6·33-s + 2.36e7·35-s + 4.95e7·36-s − 1.33e7·37-s + ⋯ |
| L(s) = 1 | − 0.634·3-s − 2·4-s − 1.70·5-s − 1.55·7-s + 1.00·8-s − 2.45·9-s − 1.61·11-s + 1.26·12-s − 2.88·13-s + 1.08·15-s + 1.00·16-s − 3.27·17-s − 2.29·19-s + 3.41·20-s + 0.988·21-s − 1.45·23-s − 0.637·24-s − 2.37·25-s + 2.21·27-s + 3.11·28-s + 0.738·29-s + 1.42·31-s − 1.73·32-s + 1.02·33-s + 2.66·35-s + 4.91·36-s − 1.16·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{7}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(23^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.27308\times 10^{7}\) |
| Root analytic conductor: | \(3.44177\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 23^{7} ,\ ( \ : [9/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( ( 1 + p^{4} T )^{7} \) |
| good | 2 | \( 1 + p^{10} T^{2} - 1455 p^{3} T^{3} + 6127 p^{7} T^{4} - 52927 p^{8} T^{5} + 424337 p^{10} T^{6} - 4631005 p^{11} T^{7} + 424337 p^{19} T^{8} - 52927 p^{26} T^{9} + 6127 p^{34} T^{10} - 1455 p^{39} T^{11} + p^{55} T^{12} + p^{63} T^{14} \) |
| 3 | \( 1 + 89 T + 18773 p T^{2} + 355648 p^{2} T^{3} + 61574636 p^{3} T^{4} + 404999824 p^{4} T^{5} + 140003108726 p^{5} T^{6} - 74633080714 p^{6} T^{7} + 140003108726 p^{14} T^{8} + 404999824 p^{22} T^{9} + 61574636 p^{30} T^{10} + 355648 p^{38} T^{11} + 18773 p^{46} T^{12} + 89 p^{54} T^{13} + p^{63} T^{14} \) | |
| 5 | \( 1 + 2388 T + 10347659 T^{2} + 15897713104 T^{3} + 41634187705269 T^{4} + 46393192014942188 T^{5} + \)\(10\!\cdots\!71\)\( T^{6} + \)\(19\!\cdots\!24\)\( p T^{7} + \)\(10\!\cdots\!71\)\( p^{9} T^{8} + 46393192014942188 p^{18} T^{9} + 41634187705269 p^{27} T^{10} + 15897713104 p^{36} T^{11} + 10347659 p^{45} T^{12} + 2388 p^{54} T^{13} + p^{63} T^{14} \) | |
| 7 | \( 1 + 9896 T + 175593517 T^{2} + 1562683196248 T^{3} + 15909705469565177 T^{4} + 16913444912014826536 p T^{5} + \)\(18\!\cdots\!25\)\( p^{2} T^{6} + \)\(16\!\cdots\!56\)\( p^{3} T^{7} + \)\(18\!\cdots\!25\)\( p^{11} T^{8} + 16913444912014826536 p^{19} T^{9} + 15909705469565177 p^{27} T^{10} + 1562683196248 p^{36} T^{11} + 175593517 p^{45} T^{12} + 9896 p^{54} T^{13} + p^{63} T^{14} \) | |
| 11 | \( 1 + 78484 T + 10901228757 T^{2} + 733893101136640 T^{3} + 58334822935416071301 T^{4} + \)\(32\!\cdots\!84\)\( T^{5} + \)\(19\!\cdots\!61\)\( T^{6} + \)\(92\!\cdots\!88\)\( T^{7} + \)\(19\!\cdots\!61\)\( p^{9} T^{8} + \)\(32\!\cdots\!84\)\( p^{18} T^{9} + 58334822935416071301 p^{27} T^{10} + 733893101136640 p^{36} T^{11} + 10901228757 p^{45} T^{12} + 78484 p^{54} T^{13} + p^{63} T^{14} \) | |
| 13 | \( 1 + 296769 T + 92822557229 T^{2} + 17981406282795684 T^{3} + \)\(32\!\cdots\!52\)\( T^{4} + \)\(35\!\cdots\!52\)\( p T^{5} + \)\(60\!\cdots\!30\)\( T^{6} + \)\(65\!\cdots\!82\)\( T^{7} + \)\(60\!\cdots\!30\)\( p^{9} T^{8} + \)\(35\!\cdots\!52\)\( p^{19} T^{9} + \)\(32\!\cdots\!52\)\( p^{27} T^{10} + 17981406282795684 p^{36} T^{11} + 92822557229 p^{45} T^{12} + 296769 p^{54} T^{13} + p^{63} T^{14} \) | |
| 17 | \( 1 + 1128820 T + 950945462119 T^{2} + 561809900824145328 T^{3} + \)\(27\!\cdots\!53\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!27\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!27\)\( p^{9} T^{8} + \)\(11\!\cdots\!16\)\( p^{18} T^{9} + \)\(27\!\cdots\!53\)\( p^{27} T^{10} + 561809900824145328 p^{36} T^{11} + 950945462119 p^{45} T^{12} + 1128820 p^{54} T^{13} + p^{63} T^{14} \) | |
| 19 | \( 1 + 1301252 T + 1939820316785 T^{2} + 1879363496695414344 T^{3} + \)\(16\!\cdots\!37\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(85\!\cdots\!09\)\( T^{6} + \)\(49\!\cdots\!44\)\( T^{7} + \)\(85\!\cdots\!09\)\( p^{9} T^{8} + \)\(12\!\cdots\!92\)\( p^{18} T^{9} + \)\(16\!\cdots\!37\)\( p^{27} T^{10} + 1879363496695414344 p^{36} T^{11} + 1939820316785 p^{45} T^{12} + 1301252 p^{54} T^{13} + p^{63} T^{14} \) | |
| 29 | \( 1 - 2813849 T + 96073558072185 T^{2} - \)\(22\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!88\)\( T^{4} - \)\(79\!\cdots\!84\)\( T^{5} + \)\(95\!\cdots\!94\)\( T^{6} - \)\(15\!\cdots\!54\)\( T^{7} + \)\(95\!\cdots\!94\)\( p^{9} T^{8} - \)\(79\!\cdots\!84\)\( p^{18} T^{9} + \)\(40\!\cdots\!88\)\( p^{27} T^{10} - \)\(22\!\cdots\!60\)\( p^{36} T^{11} + 96073558072185 p^{45} T^{12} - 2813849 p^{54} T^{13} + p^{63} T^{14} \) | |
| 31 | \( 1 - 7334751 T + 133000085478695 T^{2} - \)\(64\!\cdots\!24\)\( T^{3} + \)\(67\!\cdots\!60\)\( T^{4} - \)\(22\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!34\)\( T^{6} - \)\(57\!\cdots\!86\)\( T^{7} + \)\(20\!\cdots\!34\)\( p^{9} T^{8} - \)\(22\!\cdots\!12\)\( p^{18} T^{9} + \)\(67\!\cdots\!60\)\( p^{27} T^{10} - \)\(64\!\cdots\!24\)\( p^{36} T^{11} + 133000085478695 p^{45} T^{12} - 7334751 p^{54} T^{13} + p^{63} T^{14} \) | |
| 37 | \( 1 + 13324320 T + 509958403128099 T^{2} + \)\(69\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(76\!\cdots\!43\)\( p T^{6} + \)\(27\!\cdots\!72\)\( T^{7} + \)\(76\!\cdots\!43\)\( p^{10} T^{8} + \)\(17\!\cdots\!16\)\( p^{18} T^{9} + \)\(14\!\cdots\!41\)\( p^{27} T^{10} + \)\(69\!\cdots\!88\)\( p^{36} T^{11} + 509958403128099 p^{45} T^{12} + 13324320 p^{54} T^{13} + p^{63} T^{14} \) | |
| 41 | \( 1 + 15691573 T + 1375318214425493 T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(89\!\cdots\!84\)\( T^{4} + \)\(61\!\cdots\!04\)\( T^{5} + \)\(41\!\cdots\!78\)\( T^{6} + \)\(24\!\cdots\!58\)\( T^{7} + \)\(41\!\cdots\!78\)\( p^{9} T^{8} + \)\(61\!\cdots\!04\)\( p^{18} T^{9} + \)\(89\!\cdots\!84\)\( p^{27} T^{10} + \)\(12\!\cdots\!28\)\( p^{36} T^{11} + 1375318214425493 p^{45} T^{12} + 15691573 p^{54} T^{13} + p^{63} T^{14} \) | |
| 43 | \( 1 + 46474818 T + 2683578315052717 T^{2} + \)\(93\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!89\)\( T^{4} + \)\(96\!\cdots\!94\)\( T^{5} + \)\(27\!\cdots\!01\)\( T^{6} + \)\(60\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!01\)\( p^{9} T^{8} + \)\(96\!\cdots\!94\)\( p^{18} T^{9} + \)\(35\!\cdots\!89\)\( p^{27} T^{10} + \)\(93\!\cdots\!60\)\( p^{36} T^{11} + 2683578315052717 p^{45} T^{12} + 46474818 p^{54} T^{13} + p^{63} T^{14} \) | |
| 47 | \( 1 - 8232227 T + 3021791284626767 T^{2} - \)\(59\!\cdots\!68\)\( T^{3} + \)\(55\!\cdots\!28\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{5} + \)\(70\!\cdots\!94\)\( T^{6} - \)\(18\!\cdots\!74\)\( T^{7} + \)\(70\!\cdots\!94\)\( p^{9} T^{8} - \)\(12\!\cdots\!04\)\( p^{18} T^{9} + \)\(55\!\cdots\!28\)\( p^{27} T^{10} - \)\(59\!\cdots\!68\)\( p^{36} T^{11} + 3021791284626767 p^{45} T^{12} - 8232227 p^{54} T^{13} + p^{63} T^{14} \) | |
| 53 | \( 1 + 53545400 T + 14071117691005871 T^{2} + \)\(70\!\cdots\!08\)\( T^{3} + \)\(97\!\cdots\!53\)\( T^{4} + \)\(45\!\cdots\!64\)\( T^{5} + \)\(44\!\cdots\!55\)\( T^{6} + \)\(18\!\cdots\!28\)\( T^{7} + \)\(44\!\cdots\!55\)\( p^{9} T^{8} + \)\(45\!\cdots\!64\)\( p^{18} T^{9} + \)\(97\!\cdots\!53\)\( p^{27} T^{10} + \)\(70\!\cdots\!08\)\( p^{36} T^{11} + 14071117691005871 p^{45} T^{12} + 53545400 p^{54} T^{13} + p^{63} T^{14} \) | |
| 59 | \( 1 + 341275144 T + 87172213344057277 T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(25\!\cdots\!45\)\( T^{4} + \)\(33\!\cdots\!48\)\( T^{5} + \)\(38\!\cdots\!73\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(38\!\cdots\!73\)\( p^{9} T^{8} + \)\(33\!\cdots\!48\)\( p^{18} T^{9} + \)\(25\!\cdots\!45\)\( p^{27} T^{10} + \)\(16\!\cdots\!56\)\( p^{36} T^{11} + 87172213344057277 p^{45} T^{12} + 341275144 p^{54} T^{13} + p^{63} T^{14} \) | |
| 61 | \( 1 + 277157656 T + 81383943047092111 T^{2} + \)\(16\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!05\)\( T^{4} + \)\(43\!\cdots\!12\)\( T^{5} + \)\(56\!\cdots\!75\)\( T^{6} + \)\(64\!\cdots\!12\)\( T^{7} + \)\(56\!\cdots\!75\)\( p^{9} T^{8} + \)\(43\!\cdots\!12\)\( p^{18} T^{9} + \)\(28\!\cdots\!05\)\( p^{27} T^{10} + \)\(16\!\cdots\!44\)\( p^{36} T^{11} + 81383943047092111 p^{45} T^{12} + 277157656 p^{54} T^{13} + p^{63} T^{14} \) | |
| 67 | \( 1 - 89654580 T + 59646420871281213 T^{2} + \)\(50\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} + \)\(24\!\cdots\!52\)\( T^{5} + \)\(89\!\cdots\!09\)\( T^{6} + \)\(95\!\cdots\!76\)\( T^{7} + \)\(89\!\cdots\!09\)\( p^{9} T^{8} + \)\(24\!\cdots\!52\)\( p^{18} T^{9} + \)\(13\!\cdots\!25\)\( p^{27} T^{10} + \)\(50\!\cdots\!32\)\( p^{36} T^{11} + 59646420871281213 p^{45} T^{12} - 89654580 p^{54} T^{13} + p^{63} T^{14} \) | |
| 71 | \( 1 + 286098961 T + 51012935713112263 T^{2} - \)\(12\!\cdots\!28\)\( T^{3} - \)\(55\!\cdots\!56\)\( p T^{4} - \)\(62\!\cdots\!80\)\( T^{5} + \)\(24\!\cdots\!66\)\( T^{6} + \)\(69\!\cdots\!78\)\( T^{7} + \)\(24\!\cdots\!66\)\( p^{9} T^{8} - \)\(62\!\cdots\!80\)\( p^{18} T^{9} - \)\(55\!\cdots\!56\)\( p^{28} T^{10} - \)\(12\!\cdots\!28\)\( p^{36} T^{11} + 51012935713112263 p^{45} T^{12} + 286098961 p^{54} T^{13} + p^{63} T^{14} \) | |
| 73 | \( 1 + 637495039 T + 410258525574225253 T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!32\)\( T^{4} + \)\(19\!\cdots\!04\)\( T^{5} + \)\(58\!\cdots\!42\)\( T^{6} + \)\(14\!\cdots\!70\)\( T^{7} + \)\(58\!\cdots\!42\)\( p^{9} T^{8} + \)\(19\!\cdots\!04\)\( p^{18} T^{9} + \)\(63\!\cdots\!32\)\( p^{27} T^{10} + \)\(16\!\cdots\!20\)\( p^{36} T^{11} + 410258525574225253 p^{45} T^{12} + 637495039 p^{54} T^{13} + p^{63} T^{14} \) | |
| 79 | \( 1 - 274469546 T + 711394885574565961 T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!85\)\( T^{4} - \)\(42\!\cdots\!62\)\( T^{5} + \)\(42\!\cdots\!97\)\( T^{6} - \)\(65\!\cdots\!92\)\( T^{7} + \)\(42\!\cdots\!97\)\( p^{9} T^{8} - \)\(42\!\cdots\!62\)\( p^{18} T^{9} + \)\(22\!\cdots\!85\)\( p^{27} T^{10} - \)\(16\!\cdots\!88\)\( p^{36} T^{11} + 711394885574565961 p^{45} T^{12} - 274469546 p^{54} T^{13} + p^{63} T^{14} \) | |
| 83 | \( 1 - 1164579762 T + 1265100440875063585 T^{2} - \)\(75\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!33\)\( T^{4} - \)\(15\!\cdots\!62\)\( T^{5} + \)\(61\!\cdots\!57\)\( T^{6} - \)\(18\!\cdots\!72\)\( T^{7} + \)\(61\!\cdots\!57\)\( p^{9} T^{8} - \)\(15\!\cdots\!62\)\( p^{18} T^{9} + \)\(42\!\cdots\!33\)\( p^{27} T^{10} - \)\(75\!\cdots\!80\)\( p^{36} T^{11} + 1265100440875063585 p^{45} T^{12} - 1164579762 p^{54} T^{13} + p^{63} T^{14} \) | |
| 89 | \( 1 + 504153000 T + 1586076374292019667 T^{2} + \)\(88\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} + \)\(65\!\cdots\!24\)\( T^{5} + \)\(69\!\cdots\!51\)\( T^{6} + \)\(28\!\cdots\!44\)\( T^{7} + \)\(69\!\cdots\!51\)\( p^{9} T^{8} + \)\(65\!\cdots\!24\)\( p^{18} T^{9} + \)\(13\!\cdots\!09\)\( p^{27} T^{10} + \)\(88\!\cdots\!24\)\( p^{36} T^{11} + 1586076374292019667 p^{45} T^{12} + 504153000 p^{54} T^{13} + p^{63} T^{14} \) | |
| 97 | \( 1 + 36287928 p T + 9443697783553902375 T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(27\!\cdots\!97\)\( T^{4} + \)\(35\!\cdots\!32\)\( T^{5} + \)\(39\!\cdots\!83\)\( T^{6} + \)\(36\!\cdots\!80\)\( T^{7} + \)\(39\!\cdots\!83\)\( p^{9} T^{8} + \)\(35\!\cdots\!32\)\( p^{18} T^{9} + \)\(27\!\cdots\!97\)\( p^{27} T^{10} + \)\(17\!\cdots\!08\)\( p^{36} T^{11} + 9443697783553902375 p^{45} T^{12} + 36287928 p^{55} T^{13} + p^{63} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.240106012873132503020332058887, −8.239369191638625955044711583831, −7.77502653718426612820561321739, −7.38770665624196067152197146326, −7.26462883052541258282872541346, −6.94642512502772083637980677888, −6.60056005849286683270383785777, −6.22112520851791918325394494829, −6.17872485492719070162590457342, −6.15796151027072997250582282946, −5.66844376404208141181419048423, −5.00697785617162724963758005148, −4.99912128300218473283783112113, −4.85060682784131853203736013553, −4.60701980280803885621530720604, −4.30051483683661358373815300136, −4.23130238076880100337200484341, −4.00158639429786574385374920509, −3.33756633282684842345024010743, −3.09118033530364278605688855992, −2.89376467675347336033210175856, −2.42506267255242288549954784940, −2.24761614132626320517253327850, −1.94859496582404129657579945444, −1.47351254318745574544285578858, 0, 0, 0, 0, 0, 0, 0, 1.47351254318745574544285578858, 1.94859496582404129657579945444, 2.24761614132626320517253327850, 2.42506267255242288549954784940, 2.89376467675347336033210175856, 3.09118033530364278605688855992, 3.33756633282684842345024010743, 4.00158639429786574385374920509, 4.23130238076880100337200484341, 4.30051483683661358373815300136, 4.60701980280803885621530720604, 4.85060682784131853203736013553, 4.99912128300218473283783112113, 5.00697785617162724963758005148, 5.66844376404208141181419048423, 6.15796151027072997250582282946, 6.17872485492719070162590457342, 6.22112520851791918325394494829, 6.60056005849286683270383785777, 6.94642512502772083637980677888, 7.26462883052541258282872541346, 7.38770665624196067152197146326, 7.77502653718426612820561321739, 8.239369191638625955044711583831, 8.240106012873132503020332058887
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.