Dirichlet series
| L(s) = 1 | − 2-s + 21·3-s − 20·4-s − 35·5-s − 21·6-s − 37·7-s + 11·8-s + 252·9-s + 35·10-s − 11·11-s − 420·12-s − 133·13-s + 37·14-s − 735·15-s + 166·16-s + 21·17-s − 252·18-s − 170·19-s + 700·20-s − 777·21-s + 11·22-s − 68·23-s + 231·24-s + 700·25-s + 133·26-s + 2.26e3·27-s + 740·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 4.04·3-s − 5/2·4-s − 3.13·5-s − 1.42·6-s − 1.99·7-s + 0.486·8-s + 28/3·9-s + 1.10·10-s − 0.301·11-s − 10.1·12-s − 2.83·13-s + 0.706·14-s − 12.6·15-s + 2.59·16-s + 0.299·17-s − 3.29·18-s − 2.05·19-s + 7.82·20-s − 8.07·21-s + 0.106·22-s − 0.616·23-s + 1.96·24-s + 28/5·25-s + 1.00·26-s + 16.1·27-s + 4.99·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{7} \cdot 29^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7.33647\times 10^{9}\) |
| Root analytic conductor: | \(5.06614\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{7} \cdot 29^{7} ,\ ( \ : [3/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{7} \) |
| 5 | \( ( 1 + p T )^{7} \) | |
| 29 | \( ( 1 + p T )^{7} \) | |
| good | 2 | \( 1 + T + 21 T^{2} + 15 p T^{3} + 273 T^{4} + 551 T^{5} + 2649 T^{6} + 2681 p T^{7} + 2649 p^{3} T^{8} + 551 p^{6} T^{9} + 273 p^{9} T^{10} + 15 p^{13} T^{11} + 21 p^{15} T^{12} + p^{18} T^{13} + p^{21} T^{14} \) |
| 7 | \( 1 + 37 T + 1796 T^{2} + 47917 T^{3} + 1426006 T^{4} + 31237491 T^{5} + 715153611 T^{6} + 13089067990 T^{7} + 715153611 p^{3} T^{8} + 31237491 p^{6} T^{9} + 1426006 p^{9} T^{10} + 47917 p^{12} T^{11} + 1796 p^{15} T^{12} + 37 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 + p T + 4352 T^{2} + 66811 T^{3} + 9435666 T^{4} + 152928277 T^{5} + 1393196397 p T^{6} + 221906200330 T^{7} + 1393196397 p^{4} T^{8} + 152928277 p^{6} T^{9} + 9435666 p^{9} T^{10} + 66811 p^{12} T^{11} + 4352 p^{15} T^{12} + p^{19} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 + 133 T + 13342 T^{2} + 735693 T^{3} + 34388624 T^{4} + 849440335 T^{5} + 24454604021 T^{6} + 175666887998 T^{7} + 24454604021 p^{3} T^{8} + 849440335 p^{6} T^{9} + 34388624 p^{9} T^{10} + 735693 p^{12} T^{11} + 13342 p^{15} T^{12} + 133 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 21 T + 18662 T^{2} - 245933 T^{3} + 184095772 T^{4} - 842142119 T^{5} + 1202686535085 T^{6} - 2471410703022 T^{7} + 1202686535085 p^{3} T^{8} - 842142119 p^{6} T^{9} + 184095772 p^{9} T^{10} - 245933 p^{12} T^{11} + 18662 p^{15} T^{12} - 21 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 + 170 T + 36181 T^{2} + 3672508 T^{3} + 470183445 T^{4} + 35269408294 T^{5} + 3741257591993 T^{6} + 248193611291976 T^{7} + 3741257591993 p^{3} T^{8} + 35269408294 p^{6} T^{9} + 470183445 p^{9} T^{10} + 3672508 p^{12} T^{11} + 36181 p^{15} T^{12} + 170 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 68 T + 26757 T^{2} - 761592 T^{3} + 408422849 T^{4} - 13597110340 T^{5} + 8392192575981 T^{6} - 59485064063632 T^{7} + 8392192575981 p^{3} T^{8} - 13597110340 p^{6} T^{9} + 408422849 p^{9} T^{10} - 761592 p^{12} T^{11} + 26757 p^{15} T^{12} + 68 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 + 480 T + 214037 T^{2} + 64593680 T^{3} + 574364375 p T^{4} + 3958939665248 T^{5} + 825777844000309 T^{6} + 146607276160317280 T^{7} + 825777844000309 p^{3} T^{8} + 3958939665248 p^{6} T^{9} + 574364375 p^{10} T^{10} + 64593680 p^{12} T^{11} + 214037 p^{15} T^{12} + 480 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 1032 T + 600267 T^{2} + 255657728 T^{3} + 89823250797 T^{4} + 27447618724024 T^{5} + 7447080372004351 T^{6} + 1785040920441502720 T^{7} + 7447080372004351 p^{3} T^{8} + 27447618724024 p^{6} T^{9} + 89823250797 p^{9} T^{10} + 255657728 p^{12} T^{11} + 600267 p^{15} T^{12} + 1032 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 638 T + 447291 T^{2} + 182248732 T^{3} + 77530914417 T^{4} + 24903041218306 T^{5} + 8159072820964259 T^{6} + 52280556327244232 p T^{7} + 8159072820964259 p^{3} T^{8} + 24903041218306 p^{6} T^{9} + 77530914417 p^{9} T^{10} + 182248732 p^{12} T^{11} + 447291 p^{15} T^{12} + 638 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 512 T + 398973 T^{2} + 152623672 T^{3} + 67771792037 T^{4} + 20922169921184 T^{5} + 163937144898643 p T^{6} + 1909559682349604304 T^{7} + 163937144898643 p^{4} T^{8} + 20922169921184 p^{6} T^{9} + 67771792037 p^{9} T^{10} + 152623672 p^{12} T^{11} + 398973 p^{15} T^{12} + 512 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 111 T + 284324 T^{2} + 9903123 T^{3} + 37159928758 T^{4} + 2957669113293 T^{5} + 4587259849911315 T^{6} + 644988267769379682 T^{7} + 4587259849911315 p^{3} T^{8} + 2957669113293 p^{6} T^{9} + 37159928758 p^{9} T^{10} + 9903123 p^{12} T^{11} + 284324 p^{15} T^{12} + 111 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 410 T + 702279 T^{2} - 180724188 T^{3} + 208414492921 T^{4} - 34546039725910 T^{5} + 39512713926456295 T^{6} - 5021114037992208264 T^{7} + 39512713926456295 p^{3} T^{8} - 34546039725910 p^{6} T^{9} + 208414492921 p^{9} T^{10} - 180724188 p^{12} T^{11} + 702279 p^{15} T^{12} - 410 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 + 426 T + 819161 T^{2} + 241384580 T^{3} + 297369331833 T^{4} + 57195377783910 T^{5} + 71158241161786969 T^{6} + 10317007219357163768 T^{7} + 71158241161786969 p^{3} T^{8} + 57195377783910 p^{6} T^{9} + 297369331833 p^{9} T^{10} + 241384580 p^{12} T^{11} + 819161 p^{15} T^{12} + 426 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 + 1192 T + 1084171 T^{2} + 680888776 T^{3} + 390884697605 T^{4} + 206217200483576 T^{5} + 114049655671820439 T^{6} + 54988545004822097200 T^{7} + 114049655671820439 p^{3} T^{8} + 206217200483576 p^{6} T^{9} + 390884697605 p^{9} T^{10} + 680888776 p^{12} T^{11} + 1084171 p^{15} T^{12} + 1192 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 + 1671 T + 1835520 T^{2} + 1521190423 T^{3} + 1103345052994 T^{4} + 707389731354625 T^{5} + 433548361378690327 T^{6} + \)\(24\!\cdots\!02\)\( T^{7} + 433548361378690327 p^{3} T^{8} + 707389731354625 p^{6} T^{9} + 1103345052994 p^{9} T^{10} + 1521190423 p^{12} T^{11} + 1835520 p^{15} T^{12} + 1671 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 1324 T + 2324745 T^{2} + 2075850848 T^{3} + 2055288751325 T^{4} + 1401082298832052 T^{5} + 1037158061242619461 T^{6} + \)\(59\!\cdots\!68\)\( T^{7} + 1037158061242619461 p^{3} T^{8} + 1401082298832052 p^{6} T^{9} + 2055288751325 p^{9} T^{10} + 2075850848 p^{12} T^{11} + 2324745 p^{15} T^{12} + 1324 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 + 852 T + 735871 T^{2} + 590922536 T^{3} + 348127313285 T^{4} + 211229298550508 T^{5} + 141403326646770155 T^{6} + 61508593982526659760 T^{7} + 141403326646770155 p^{3} T^{8} + 211229298550508 p^{6} T^{9} + 348127313285 p^{9} T^{10} + 590922536 p^{12} T^{11} + 735871 p^{15} T^{12} + 852 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 366 T + 3107369 T^{2} - 1038966900 T^{3} + 4278026594629 T^{4} - 1250662865862098 T^{5} + 3389706483803464701 T^{6} - \)\(81\!\cdots\!92\)\( T^{7} + 3389706483803464701 p^{3} T^{8} - 1250662865862098 p^{6} T^{9} + 4278026594629 p^{9} T^{10} - 1038966900 p^{12} T^{11} + 3107369 p^{15} T^{12} - 366 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 470 T + 2469149 T^{2} - 1396541012 T^{3} + 3118284728717 T^{4} - 1789356214405850 T^{5} + 2570451621538769289 T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + 2570451621538769289 p^{3} T^{8} - 1789356214405850 p^{6} T^{9} + 3118284728717 p^{9} T^{10} - 1396541012 p^{12} T^{11} + 2469149 p^{15} T^{12} - 470 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 - 51 T + 3715466 T^{2} - 236688179 T^{3} + 6469528959684 T^{4} - 397638706562113 T^{5} + 6897839542444140921 T^{6} - \)\(36\!\cdots\!74\)\( T^{7} + 6897839542444140921 p^{3} T^{8} - 397638706562113 p^{6} T^{9} + 6469528959684 p^{9} T^{10} - 236688179 p^{12} T^{11} + 3715466 p^{15} T^{12} - 51 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 3322 T + 9387531 T^{2} + 17302290396 T^{3} + 28599963295321 T^{4} + 37191184944922886 T^{5} + 44499840183929857371 T^{6} + \)\(44\!\cdots\!12\)\( T^{7} + 44499840183929857371 p^{3} T^{8} + 37191184944922886 p^{6} T^{9} + 28599963295321 p^{9} T^{10} + 17302290396 p^{12} T^{11} + 9387531 p^{15} T^{12} + 3322 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−5.23235255174950081729141450686, −5.20073945407256993074817547396, −5.15550088687433650294532513695, −5.05820479491504170794441593692, −4.54174264830510166633204995829, −4.43643196139673709859606233206, −4.30446393105340264804911241381, −4.30131584304351238418513326503, −4.11616575816282889043831038620, −3.96178302936193827173180502423, −3.89612041912895302225633650716, −3.47498776019233954827415395157, −3.38457364206971820234017084145, −3.37804862897951705431615827181, −3.30349003967328716598194227499, −3.27023897586208479596273479086, −2.80681374297313137501342389484, −2.79997204150534965615644183962, −2.50698734870844176388196730566, −2.41159589457100068617355794123, −2.05910758858564424250780288909, −1.57889007058541081069697262654, −1.54643358343931661440859537037, −1.51781887907631968426330964140, −1.38253862220376071129854154992, 0, 0, 0, 0, 0, 0, 0, 1.38253862220376071129854154992, 1.51781887907631968426330964140, 1.54643358343931661440859537037, 1.57889007058541081069697262654, 2.05910758858564424250780288909, 2.41159589457100068617355794123, 2.50698734870844176388196730566, 2.79997204150534965615644183962, 2.80681374297313137501342389484, 3.27023897586208479596273479086, 3.30349003967328716598194227499, 3.37804862897951705431615827181, 3.38457364206971820234017084145, 3.47498776019233954827415395157, 3.89612041912895302225633650716, 3.96178302936193827173180502423, 4.11616575816282889043831038620, 4.30131584304351238418513326503, 4.30446393105340264804911241381, 4.43643196139673709859606233206, 4.54174264830510166633204995829, 5.05820479491504170794441593692, 5.15550088687433650294532513695, 5.20073945407256993074817547396, 5.23235255174950081729141450686
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.