Dirichlet series
| L(s) = 1 | − 2·2-s − 21·3-s − 15·4-s + 35·5-s + 42·6-s − 50·7-s + 27·8-s + 252·9-s − 70·10-s + 76·11-s + 315·12-s + 30·13-s + 100·14-s − 735·15-s + 113·16-s − 140·17-s − 504·18-s + 90·19-s − 525·20-s + 1.05e3·21-s − 152·22-s + 34·23-s − 567·24-s + 700·25-s − 60·26-s − 2.26e3·27-s + 750·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4.04·3-s − 1.87·4-s + 3.13·5-s + 2.85·6-s − 2.69·7-s + 1.19·8-s + 28/3·9-s − 2.21·10-s + 2.08·11-s + 7.57·12-s + 0.640·13-s + 1.90·14-s − 12.6·15-s + 1.76·16-s − 1.99·17-s − 6.59·18-s + 1.08·19-s − 5.86·20-s + 10.9·21-s − 1.47·22-s + 0.308·23-s − 4.82·24-s + 28/5·25-s − 0.452·26-s − 16.1·27-s + 5.06·28-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: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{7} \cdot 29^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(2.753178948\) |
| \(L(\frac12)\) | \(\approx\) | \(2.753178948\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 + p T + 19 T^{2} + 41 T^{3} + 25 p^{3} T^{4} + 387 T^{5} + 185 p^{3} T^{6} + 1579 p T^{7} + 185 p^{6} T^{8} + 387 p^{6} T^{9} + 25 p^{12} T^{10} + 41 p^{12} T^{11} + 19 p^{15} T^{12} + p^{19} T^{13} + p^{21} T^{14} \) |
| 7 | \( 1 + 50 T + 2166 T^{2} + 62472 T^{3} + 1814424 T^{4} + 41474014 T^{5} + 935177403 T^{6} + 17274887696 T^{7} + 935177403 p^{3} T^{8} + 41474014 p^{6} T^{9} + 1814424 p^{9} T^{10} + 62472 p^{12} T^{11} + 2166 p^{15} T^{12} + 50 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 76 T + 6531 T^{2} - 341908 T^{3} + 15932616 T^{4} - 627746660 T^{5} + 22484848974 T^{6} - 817707900184 T^{7} + 22484848974 p^{3} T^{8} - 627746660 p^{6} T^{9} + 15932616 p^{9} T^{10} - 341908 p^{12} T^{11} + 6531 p^{15} T^{12} - 76 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 30 T + 5430 T^{2} - 172702 T^{3} + 19561936 T^{4} - 353786730 T^{5} + 45086623549 T^{6} - 854392315828 T^{7} + 45086623549 p^{3} T^{8} - 353786730 p^{6} T^{9} + 19561936 p^{9} T^{10} - 172702 p^{12} T^{11} + 5430 p^{15} T^{12} - 30 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 + 140 T + 21616 T^{2} + 1595902 T^{3} + 151928862 T^{4} + 9191156404 T^{5} + 861332252673 T^{6} + 51102907071364 T^{7} + 861332252673 p^{3} T^{8} + 9191156404 p^{6} T^{9} + 151928862 p^{9} T^{10} + 1595902 p^{12} T^{11} + 21616 p^{15} T^{12} + 140 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 - 90 T + 18469 T^{2} - 31764 p T^{3} + 130314373 T^{4} - 608196854 T^{5} + 1048759228569 T^{6} - 10968510092744 T^{7} + 1048759228569 p^{3} T^{8} - 608196854 p^{6} T^{9} + 130314373 p^{9} T^{10} - 31764 p^{13} T^{11} + 18469 p^{15} T^{12} - 90 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 - 34 T + 57522 T^{2} - 601396 T^{3} + 62732760 p T^{4} + 14721072002 T^{5} + 22678896398095 T^{6} + 413738788211176 T^{7} + 22678896398095 p^{3} T^{8} + 14721072002 p^{6} T^{9} + 62732760 p^{10} T^{10} - 601396 p^{12} T^{11} + 57522 p^{15} T^{12} - 34 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 524 T + 216457 T^{2} - 67046168 T^{3} + 17937386149 T^{4} - 4108601336372 T^{5} + 848529682882461 T^{6} - 153293547374151504 T^{7} + 848529682882461 p^{3} T^{8} - 4108601336372 p^{6} T^{9} + 17937386149 p^{9} T^{10} - 67046168 p^{12} T^{11} + 216457 p^{15} T^{12} - 524 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 28 T + 87372 T^{2} - 3284206 T^{3} + 6985543210 T^{4} - 284555878876 T^{5} + 428170938097933 T^{6} - 24978262328385412 T^{7} + 428170938097933 p^{3} T^{8} - 284555878876 p^{6} T^{9} + 6985543210 p^{9} T^{10} - 3284206 p^{12} T^{11} + 87372 p^{15} T^{12} + 28 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 - 1532 T + 1436660 T^{2} - 943584750 T^{3} + 484684765182 T^{4} - 200227115787372 T^{5} + 68667684339148117 T^{6} - 19644164023038905876 T^{7} + 68667684339148117 p^{3} T^{8} - 200227115787372 p^{6} T^{9} + 484684765182 p^{9} T^{10} - 943584750 p^{12} T^{11} + 1436660 p^{15} T^{12} - 1532 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 + 464 T + 239094 T^{2} + 59377006 T^{3} + 22371054904 T^{4} + 3273282215368 T^{5} + 1364956108196167 T^{6} + 183555450516418132 T^{7} + 1364956108196167 p^{3} T^{8} + 3273282215368 p^{6} T^{9} + 22371054904 p^{9} T^{10} + 59377006 p^{12} T^{11} + 239094 p^{15} T^{12} + 464 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 360 T + 444364 T^{2} + 118324884 T^{3} + 90595740690 T^{4} + 18884889539688 T^{5} + 11978054563535223 T^{6} + 2129858244743563320 T^{7} + 11978054563535223 p^{3} T^{8} + 18884889539688 p^{6} T^{9} + 90595740690 p^{9} T^{10} + 118324884 p^{12} T^{11} + 444364 p^{15} T^{12} + 360 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 282 T + 414832 T^{2} - 158982736 T^{3} + 125462767638 T^{4} - 41123608910334 T^{5} + 24827057402722909 T^{6} - 7899927468616729168 T^{7} + 24827057402722909 p^{3} T^{8} - 41123608910334 p^{6} T^{9} + 125462767638 p^{9} T^{10} - 158982736 p^{12} T^{11} + 414832 p^{15} T^{12} - 282 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 766 T + 1199681 T^{2} - 621550764 T^{3} + 594571980177 T^{4} - 242795862037170 T^{5} + 181128249674429929 T^{6} - 61075511445387462568 T^{7} + 181128249674429929 p^{3} T^{8} - 242795862037170 p^{6} T^{9} + 594571980177 p^{9} T^{10} - 621550764 p^{12} T^{11} + 1199681 p^{15} T^{12} - 766 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 1200 T + 1557951 T^{2} - 1164433544 T^{3} + 940287529001 T^{4} - 544237158673744 T^{5} + 335148051418867919 T^{6} - \)\(15\!\cdots\!96\)\( T^{7} + 335148051418867919 p^{3} T^{8} - 544237158673744 p^{6} T^{9} + 940287529001 p^{9} T^{10} - 1164433544 p^{12} T^{11} + 1557951 p^{15} T^{12} - 1200 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 - 1546 T + 2405590 T^{2} - 2274055692 T^{3} + 2115500278656 T^{4} - 1475089660569462 T^{5} + 1019032233662177143 T^{6} - \)\(56\!\cdots\!16\)\( T^{7} + 1019032233662177143 p^{3} T^{8} - 1475089660569462 p^{6} T^{9} + 2115500278656 p^{9} T^{10} - 2274055692 p^{12} T^{11} + 2405590 p^{15} T^{12} - 1546 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 - 1802 T + 2611041 T^{2} - 2499303500 T^{3} + 2276927359413 T^{4} - 1700113447871062 T^{5} + 1248678279381678405 T^{6} - \)\(76\!\cdots\!20\)\( T^{7} + 1248678279381678405 p^{3} T^{8} - 1700113447871062 p^{6} T^{9} + 2276927359413 p^{9} T^{10} - 2499303500 p^{12} T^{11} + 2611041 p^{15} T^{12} - 1802 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 + 220 T + 1197248 T^{2} + 201740502 T^{3} + 826865904782 T^{4} + 53635991714724 T^{5} + 387121544848050585 T^{6} + 15392875431711568148 T^{7} + 387121544848050585 p^{3} T^{8} + 53635991714724 p^{6} T^{9} + 826865904782 p^{9} T^{10} + 201740502 p^{12} T^{11} + 1197248 p^{15} T^{12} + 220 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 1298 T + 1401925 T^{2} - 1067815236 T^{3} + 876448151817 T^{4} - 361591908143502 T^{5} + 224605827529722725 T^{6} - 98772179358995163832 T^{7} + 224605827529722725 p^{3} T^{8} - 361591908143502 p^{6} T^{9} + 876448151817 p^{9} T^{10} - 1067815236 p^{12} T^{11} + 1401925 p^{15} T^{12} - 1298 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 - 1652 T + 2946286 T^{2} - 2600702126 T^{3} + 2911070520872 T^{4} - 2065211564943492 T^{5} + 2149590463708626967 T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + 2149590463708626967 p^{3} T^{8} - 2065211564943492 p^{6} T^{9} + 2911070520872 p^{9} T^{10} - 2600702126 p^{12} T^{11} + 2946286 p^{15} T^{12} - 1652 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 - 2846 T + 5239030 T^{2} - 6642226306 T^{3} + 7422551878344 T^{4} - 7434492428937098 T^{5} + 7313624151148701705 T^{6} - 71720768886987270412 p T^{7} + 7313624151148701705 p^{3} T^{8} - 7434492428937098 p^{6} T^{9} + 7422551878344 p^{9} T^{10} - 6642226306 p^{12} T^{11} + 5239030 p^{15} T^{12} - 2846 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 - 1110 T + 4322564 T^{2} - 2845349012 T^{3} + 7614623802234 T^{4} - 3022963837913410 T^{5} + 8540678244731001633 T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + 8540678244731001633 p^{3} T^{8} - 3022963837913410 p^{6} T^{9} + 7614623802234 p^{9} T^{10} - 2845349012 p^{12} T^{11} + 4322564 p^{15} T^{12} - 1110 p^{18} T^{13} + p^{21} T^{14} \) | |
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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
−5.01200174110809291543161844425, −5.00982384976872826631653443736, −4.86405504162719696594876375774, −4.74269334162501638398106232719, −4.32976011783828265162166917631, −4.10687724538625121853682746059, −3.96020834428211944202156491478, −3.84962968502172973922307146854, −3.81995828003394997278828097777, −3.75858753725271660509512778420, −3.29659420375294180500554826932, −2.97281543606151884408825002353, −2.81728917315718518384932292266, −2.54374036600031587190696941934, −2.47698213947610411168312872873, −2.19754132359959355602797592704, −1.91878660262277380461491291636, −1.64845617539078381701702313123, −1.39753963967844542091092476708, −1.00545073043389954214883431763, −0.880889658107417387830054797693, −0.808166535542324138544707266570, −0.58552384414619822930587158013, −0.53059391121560305721812868442, −0.41648380067357538205806250363, 0.41648380067357538205806250363, 0.53059391121560305721812868442, 0.58552384414619822930587158013, 0.808166535542324138544707266570, 0.880889658107417387830054797693, 1.00545073043389954214883431763, 1.39753963967844542091092476708, 1.64845617539078381701702313123, 1.91878660262277380461491291636, 2.19754132359959355602797592704, 2.47698213947610411168312872873, 2.54374036600031587190696941934, 2.81728917315718518384932292266, 2.97281543606151884408825002353, 3.29659420375294180500554826932, 3.75858753725271660509512778420, 3.81995828003394997278828097777, 3.84962968502172973922307146854, 3.96020834428211944202156491478, 4.10687724538625121853682746059, 4.32976011783828265162166917631, 4.74269334162501638398106232719, 4.86405504162719696594876375774, 5.00982384976872826631653443736, 5.01200174110809291543161844425
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.