Dirichlet series
| L(s) = 1 | − 8·2-s − 47·3-s − 27·4-s − 64·5-s + 376·6-s − 293·7-s + 314·8-s + 400·9-s + 512·10-s − 1.45e3·11-s + 1.26e3·12-s − 536·13-s + 2.34e3·14-s + 3.00e3·15-s + 435·16-s − 3.06e3·17-s − 3.20e3·18-s − 1.90e3·19-s + 1.72e3·20-s + 1.37e4·21-s + 1.16e4·22-s − 3.98e3·23-s − 1.47e4·24-s − 2.77e3·25-s + 4.28e3·26-s + 1.60e4·27-s + 7.91e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3.01·3-s − 0.843·4-s − 1.14·5-s + 4.26·6-s − 2.26·7-s + 1.73·8-s + 1.64·9-s + 1.61·10-s − 3.63·11-s + 2.54·12-s − 0.879·13-s + 3.19·14-s + 3.45·15-s + 0.424·16-s − 2.57·17-s − 2.32·18-s − 1.20·19-s + 0.965·20-s + 6.81·21-s + 5.13·22-s − 1.57·23-s − 5.22·24-s − 0.887·25-s + 1.24·26-s + 4.23·27-s + 1.90·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{7}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(37^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(259140.\) |
| Root analytic conductor: | \(2.43602\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 37^{7} ,\ ( \ : [5/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(3)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 37 | \( ( 1 + p^{2} T )^{7} \) |
| good | 2 | \( 1 + p^{3} T + 91 T^{2} + 315 p T^{3} + 2275 p T^{4} + 3355 p^{3} T^{5} + 19917 p^{3} T^{6} + 26319 p^{5} T^{7} + 19917 p^{8} T^{8} + 3355 p^{13} T^{9} + 2275 p^{16} T^{10} + 315 p^{21} T^{11} + 91 p^{25} T^{12} + p^{33} T^{13} + p^{35} T^{14} \) |
| 3 | \( 1 + 47 T + 67 p^{3} T^{2} + 50174 T^{3} + 1176889 T^{4} + 7876703 p T^{5} + 141813452 p T^{6} + 255842851 p^{3} T^{7} + 141813452 p^{6} T^{8} + 7876703 p^{11} T^{9} + 1176889 p^{15} T^{10} + 50174 p^{20} T^{11} + 67 p^{28} T^{12} + 47 p^{30} T^{13} + p^{35} T^{14} \) | |
| 5 | \( 1 + 64 T + 1374 p T^{2} + 13468 p^{2} T^{3} + 4917549 p T^{4} + 1238804546 T^{5} + 97746066184 T^{6} + 948330630148 p T^{7} + 97746066184 p^{5} T^{8} + 1238804546 p^{10} T^{9} + 4917549 p^{16} T^{10} + 13468 p^{22} T^{11} + 1374 p^{26} T^{12} + 64 p^{30} T^{13} + p^{35} T^{14} \) | |
| 7 | \( 1 + 293 T + 16620 p T^{2} + 22158713 T^{3} + 4929426770 T^{4} + 699628755547 T^{5} + 116419691757575 T^{6} + 1967406663448594 p T^{7} + 116419691757575 p^{5} T^{8} + 699628755547 p^{10} T^{9} + 4929426770 p^{15} T^{10} + 22158713 p^{20} T^{11} + 16620 p^{26} T^{12} + 293 p^{30} T^{13} + p^{35} T^{14} \) | |
| 11 | \( 1 + 1457 T + 1577309 T^{2} + 1175006518 T^{3} + 740050917761 T^{4} + 34741050664877 p T^{5} + 179387852123281792 T^{6} + 74236782526173050207 T^{7} + 179387852123281792 p^{5} T^{8} + 34741050664877 p^{11} T^{9} + 740050917761 p^{15} T^{10} + 1175006518 p^{20} T^{11} + 1577309 p^{25} T^{12} + 1457 p^{30} T^{13} + p^{35} T^{14} \) | |
| 13 | \( 1 + 536 T + 1127102 T^{2} + 357975092 T^{3} + 608374980745 T^{4} + 165939394147042 T^{5} + 285413685852246936 T^{6} + 78791722113763925716 T^{7} + 285413685852246936 p^{5} T^{8} + 165939394147042 p^{10} T^{9} + 608374980745 p^{15} T^{10} + 357975092 p^{20} T^{11} + 1127102 p^{25} T^{12} + 536 p^{30} T^{13} + p^{35} T^{14} \) | |
| 17 | \( 1 + 3068 T + 9323183 T^{2} + 14143415112 T^{3} + 20891571621149 T^{4} + 14387970191267460 T^{5} + 12273787511156491987 T^{6} + \)\(11\!\cdots\!44\)\( T^{7} + 12273787511156491987 p^{5} T^{8} + 14387970191267460 p^{10} T^{9} + 20891571621149 p^{15} T^{10} + 14143415112 p^{20} T^{11} + 9323183 p^{25} T^{12} + 3068 p^{30} T^{13} + p^{35} T^{14} \) | |
| 19 | \( 1 + 100 p T + 12368813 T^{2} + 16578669432 T^{3} + 69251640290421 T^{4} + 76180003149710388 T^{5} + \)\(25\!\cdots\!49\)\( T^{6} + \)\(23\!\cdots\!48\)\( T^{7} + \)\(25\!\cdots\!49\)\( p^{5} T^{8} + 76180003149710388 p^{10} T^{9} + 69251640290421 p^{15} T^{10} + 16578669432 p^{20} T^{11} + 12368813 p^{25} T^{12} + 100 p^{31} T^{13} + p^{35} T^{14} \) | |
| 23 | \( 1 + 3986 T + 24913578 T^{2} + 56323103550 T^{3} + 219736382659765 T^{4} + 305253478359126668 T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!80\)\( p^{5} T^{8} + 305253478359126668 p^{10} T^{9} + 219736382659765 p^{15} T^{10} + 56323103550 p^{20} T^{11} + 24913578 p^{25} T^{12} + 3986 p^{30} T^{13} + p^{35} T^{14} \) | |
| 29 | \( 1 + 7436 T + 59455110 T^{2} + 99115079336 T^{3} + 626182848385577 T^{4} + 1013936052885791410 T^{5} + \)\(37\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{7} + \)\(37\!\cdots\!52\)\( p^{5} T^{8} + 1013936052885791410 p^{10} T^{9} + 626182848385577 p^{15} T^{10} + 99115079336 p^{20} T^{11} + 59455110 p^{25} T^{12} + 7436 p^{30} T^{13} + p^{35} T^{14} \) | |
| 31 | \( 1 - 5776 T + 164424378 T^{2} - 717173439400 T^{3} + 11790781294261793 T^{4} - 40732674798185768252 T^{5} + \)\(50\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!80\)\( p^{5} T^{8} - 40732674798185768252 p^{10} T^{9} + 11790781294261793 p^{15} T^{10} - 717173439400 p^{20} T^{11} + 164424378 p^{25} T^{12} - 5776 p^{30} T^{13} + p^{35} T^{14} \) | |
| 41 | \( 1 + 25089 T + 906412619 T^{2} + 15490917460504 T^{3} + 321156632090362235 T^{4} + \)\(41\!\cdots\!55\)\( T^{5} + \)\(61\!\cdots\!66\)\( T^{6} + \)\(62\!\cdots\!25\)\( T^{7} + \)\(61\!\cdots\!66\)\( p^{5} T^{8} + \)\(41\!\cdots\!55\)\( p^{10} T^{9} + 321156632090362235 p^{15} T^{10} + 15490917460504 p^{20} T^{11} + 906412619 p^{25} T^{12} + 25089 p^{30} T^{13} + p^{35} T^{14} \) | |
| 43 | \( 1 + 22538 T + 858580973 T^{2} + 14175971919636 T^{3} + 313066026990233021 T^{4} + \)\(40\!\cdots\!18\)\( T^{5} + \)\(67\!\cdots\!01\)\( T^{6} + \)\(17\!\cdots\!92\)\( p T^{7} + \)\(67\!\cdots\!01\)\( p^{5} T^{8} + \)\(40\!\cdots\!18\)\( p^{10} T^{9} + 313066026990233021 p^{15} T^{10} + 14175971919636 p^{20} T^{11} + 858580973 p^{25} T^{12} + 22538 p^{30} T^{13} + p^{35} T^{14} \) | |
| 47 | \( 1 + 60861 T + 2901130908 T^{2} + 95363792135089 T^{3} + 2632544785781789546 T^{4} + \)\(58\!\cdots\!87\)\( T^{5} + \)\(11\!\cdots\!03\)\( T^{6} + \)\(18\!\cdots\!66\)\( T^{7} + \)\(11\!\cdots\!03\)\( p^{5} T^{8} + \)\(58\!\cdots\!87\)\( p^{10} T^{9} + 2632544785781789546 p^{15} T^{10} + 95363792135089 p^{20} T^{11} + 2901130908 p^{25} T^{12} + 60861 p^{30} T^{13} + p^{35} T^{14} \) | |
| 53 | \( 1 + 15681 T + 1437228358 T^{2} + 250860236213 p T^{3} + 992527215222452096 T^{4} + \)\(49\!\cdots\!95\)\( T^{5} + \)\(49\!\cdots\!09\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{7} + \)\(49\!\cdots\!09\)\( p^{5} T^{8} + \)\(49\!\cdots\!95\)\( p^{10} T^{9} + 992527215222452096 p^{15} T^{10} + 250860236213 p^{21} T^{11} + 1437228358 p^{25} T^{12} + 15681 p^{30} T^{13} + p^{35} T^{14} \) | |
| 59 | \( 1 + 54536 T + 3903678129 T^{2} + 169451456470960 T^{3} + 7261803770835657081 T^{4} + \)\(25\!\cdots\!24\)\( T^{5} + \)\(80\!\cdots\!21\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(80\!\cdots\!21\)\( p^{5} T^{8} + \)\(25\!\cdots\!24\)\( p^{10} T^{9} + 7261803770835657081 p^{15} T^{10} + 169451456470960 p^{20} T^{11} + 3903678129 p^{25} T^{12} + 54536 p^{30} T^{13} + p^{35} T^{14} \) | |
| 61 | \( 1 - 48694 T + 4485013102 T^{2} - 131671964831002 T^{3} + 6734921850113642625 T^{4} - \)\(11\!\cdots\!90\)\( T^{5} + \)\(52\!\cdots\!04\)\( T^{6} - \)\(66\!\cdots\!24\)\( T^{7} + \)\(52\!\cdots\!04\)\( p^{5} T^{8} - \)\(11\!\cdots\!90\)\( p^{10} T^{9} + 6734921850113642625 p^{15} T^{10} - 131671964831002 p^{20} T^{11} + 4485013102 p^{25} T^{12} - 48694 p^{30} T^{13} + p^{35} T^{14} \) | |
| 67 | \( 1 + 39724 T + 5103294402 T^{2} + 249396680075460 T^{3} + 15042713139824200969 T^{4} + \)\(64\!\cdots\!44\)\( T^{5} + \)\(31\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(31\!\cdots\!20\)\( p^{5} T^{8} + \)\(64\!\cdots\!44\)\( p^{10} T^{9} + 15042713139824200969 p^{15} T^{10} + 249396680075460 p^{20} T^{11} + 5103294402 p^{25} T^{12} + 39724 p^{30} T^{13} + p^{35} T^{14} \) | |
| 71 | \( 1 + 92187 T + 12679023680 T^{2} + 829507162716595 T^{3} + 65375640077222398010 T^{4} + \)\(32\!\cdots\!53\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} + \)\(75\!\cdots\!90\)\( T^{7} + \)\(18\!\cdots\!87\)\( p^{5} T^{8} + \)\(32\!\cdots\!53\)\( p^{10} T^{9} + 65375640077222398010 p^{15} T^{10} + 829507162716595 p^{20} T^{11} + 12679023680 p^{25} T^{12} + 92187 p^{30} T^{13} + p^{35} T^{14} \) | |
| 73 | \( 1 - 73251 T + 9532543679 T^{2} - 545735355745416 T^{3} + 40063000863915235791 T^{4} - \)\(26\!\cdots\!45\)\( p T^{5} + \)\(10\!\cdots\!30\)\( T^{6} - \)\(44\!\cdots\!11\)\( T^{7} + \)\(10\!\cdots\!30\)\( p^{5} T^{8} - \)\(26\!\cdots\!45\)\( p^{11} T^{9} + 40063000863915235791 p^{15} T^{10} - 545735355745416 p^{20} T^{11} + 9532543679 p^{25} T^{12} - 73251 p^{30} T^{13} + p^{35} T^{14} \) | |
| 79 | \( 1 - 78604 T + 8038670990 T^{2} - 465456169662348 T^{3} + 51930561645791714477 T^{4} - \)\(30\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(91\!\cdots\!20\)\( T^{7} + \)\(21\!\cdots\!00\)\( p^{5} T^{8} - \)\(30\!\cdots\!48\)\( p^{10} T^{9} + 51930561645791714477 p^{15} T^{10} - 465456169662348 p^{20} T^{11} + 8038670990 p^{25} T^{12} - 78604 p^{30} T^{13} + p^{35} T^{14} \) | |
| 83 | \( 1 + 82223 T + 19477725828 T^{2} + 1395273312303567 T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!73\)\( T^{5} + \)\(11\!\cdots\!23\)\( T^{6} + \)\(55\!\cdots\!66\)\( T^{7} + \)\(11\!\cdots\!23\)\( p^{5} T^{8} + \)\(11\!\cdots\!73\)\( p^{10} T^{9} + \)\(18\!\cdots\!66\)\( p^{15} T^{10} + 1395273312303567 p^{20} T^{11} + 19477725828 p^{25} T^{12} + 82223 p^{30} T^{13} + p^{35} T^{14} \) | |
| 89 | \( 1 - 181680 T + 36788489807 T^{2} - 4377948660616608 T^{3} + \)\(53\!\cdots\!05\)\( T^{4} - \)\(49\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!67\)\( T^{6} - \)\(34\!\cdots\!72\)\( T^{7} + \)\(46\!\cdots\!67\)\( p^{5} T^{8} - \)\(49\!\cdots\!00\)\( p^{10} T^{9} + \)\(53\!\cdots\!05\)\( p^{15} T^{10} - 4377948660616608 p^{20} T^{11} + 36788489807 p^{25} T^{12} - 181680 p^{30} T^{13} + p^{35} T^{14} \) | |
| 97 | \( 1 - 39092 T + 17868975503 T^{2} - 1831866441048984 T^{3} + \)\(26\!\cdots\!21\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(30\!\cdots\!15\)\( T^{6} - \)\(27\!\cdots\!04\)\( T^{7} + \)\(30\!\cdots\!15\)\( p^{5} T^{8} - \)\(26\!\cdots\!36\)\( p^{10} T^{9} + \)\(26\!\cdots\!21\)\( p^{15} T^{10} - 1831866441048984 p^{20} T^{11} + 17868975503 p^{25} T^{12} - 39092 p^{30} T^{13} + p^{35} T^{14} \) | |
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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
−8.346862539298897906404730208073, −8.272311634083730274581349465153, −8.145684904248472965429304161066, −7.64754677415056570613600658560, −7.45999864491292780552190392363, −7.43266112381139355472735746667, −6.72439166459398833108153188078, −6.51112998817414404519069734727, −6.46405675238850269031580193616, −6.25759601606475927393856056544, −6.07644226310745780541732569965, −6.07400461097015585447830508583, −5.31622936116917695150930153622, −5.27438630646345199628050475351, −5.18047566977822567525060939769, −4.85913510688071767076347182400, −4.85153325815741329613424435140, −4.50441968417201622212036209815, −3.93481134807616862868337659004, −3.42722617813013955449021377342, −3.31757310784119630656231181565, −3.13574647405621265028183028834, −2.52814137503445925231116698277, −2.19920073146592356759519880610, −1.72269788420501144990646336257, 0, 0, 0, 0, 0, 0, 0, 1.72269788420501144990646336257, 2.19920073146592356759519880610, 2.52814137503445925231116698277, 3.13574647405621265028183028834, 3.31757310784119630656231181565, 3.42722617813013955449021377342, 3.93481134807616862868337659004, 4.50441968417201622212036209815, 4.85153325815741329613424435140, 4.85913510688071767076347182400, 5.18047566977822567525060939769, 5.27438630646345199628050475351, 5.31622936116917695150930153622, 6.07400461097015585447830508583, 6.07644226310745780541732569965, 6.25759601606475927393856056544, 6.46405675238850269031580193616, 6.51112998817414404519069734727, 6.72439166459398833108153188078, 7.43266112381139355472735746667, 7.45999864491292780552190392363, 7.64754677415056570613600658560, 8.145684904248472965429304161066, 8.272311634083730274581349465153, 8.346862539298897906404730208073
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.