Dirichlet series
| L(s) = 1 | + 7·2-s + 28·4-s − 5·5-s + 7·7-s + 84·8-s − 35·10-s − 4·11-s + 14·13-s + 49·14-s + 210·16-s − 4·17-s + 7·19-s − 140·20-s − 28·22-s + 23-s + 2·25-s + 98·26-s + 196·28-s − 3·29-s + 8·31-s + 462·32-s − 28·34-s − 35·35-s + 18·37-s + 49·38-s − 420·40-s − 3·41-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 14·4-s − 2.23·5-s + 2.64·7-s + 29.6·8-s − 11.0·10-s − 1.20·11-s + 3.88·13-s + 13.0·14-s + 52.5·16-s − 0.970·17-s + 1.60·19-s − 31.3·20-s − 5.96·22-s + 0.208·23-s + 2/5·25-s + 19.2·26-s + 37.0·28-s − 0.557·29-s + 1.43·31-s + 81.6·32-s − 4.80·34-s − 5.91·35-s + 2.95·37-s + 7.94·38-s − 66.4·40-s − 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{14} \cdot 241^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.98365\times 10^{10}\) |
| Root analytic conductor: | \(5.88549\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{14} \cdot 241^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1526.547354\) |
| \(L(\frac12)\) | \(\approx\) | \(1526.547354\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 3 | \( 1 \) | |
| 241 | \( ( 1 - T )^{7} \) | |
| good | 5 | \( 1 + p T + 23 T^{2} + 78 T^{3} + 257 T^{4} + 711 T^{5} + 371 p T^{6} + 4172 T^{7} + 371 p^{2} T^{8} + 711 p^{2} T^{9} + 257 p^{3} T^{10} + 78 p^{4} T^{11} + 23 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - p T + 39 T^{2} - 144 T^{3} + 519 T^{4} - 1649 T^{5} + 759 p T^{6} - 14608 T^{7} + 759 p^{2} T^{8} - 1649 p^{2} T^{9} + 519 p^{3} T^{10} - 144 p^{4} T^{11} + 39 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 4 T + 23 T^{2} + 8 p T^{3} + 387 T^{4} + 956 T^{5} + 3917 T^{6} + 11280 T^{7} + 3917 p T^{8} + 956 p^{2} T^{9} + 387 p^{3} T^{10} + 8 p^{5} T^{11} + 23 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 14 T + 99 T^{2} - 514 T^{3} + 2459 T^{4} - 11062 T^{5} + 3385 p T^{6} - 160996 T^{7} + 3385 p^{2} T^{8} - 11062 p^{2} T^{9} + 2459 p^{3} T^{10} - 514 p^{4} T^{11} + 99 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 4 T + 59 T^{2} + 218 T^{3} + 1891 T^{4} + 6232 T^{5} + 43477 T^{6} + 125588 T^{7} + 43477 p T^{8} + 6232 p^{2} T^{9} + 1891 p^{3} T^{10} + 218 p^{4} T^{11} + 59 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 7 T + 73 T^{2} - 406 T^{3} + 2689 T^{4} - 11719 T^{5} + 63545 T^{6} - 248192 T^{7} + 63545 p T^{8} - 11719 p^{2} T^{9} + 2689 p^{3} T^{10} - 406 p^{4} T^{11} + 73 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - T + 87 T^{2} - 58 T^{3} + 2855 T^{4} - 761 T^{5} + 51905 T^{6} + 2120 T^{7} + 51905 p T^{8} - 761 p^{2} T^{9} + 2855 p^{3} T^{10} - 58 p^{4} T^{11} + 87 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 3 T + 119 T^{2} + 210 T^{3} + 6905 T^{4} + 8085 T^{5} + 274263 T^{6} + 259228 T^{7} + 274263 p T^{8} + 8085 p^{2} T^{9} + 6905 p^{3} T^{10} + 210 p^{4} T^{11} + 119 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 8 T + 151 T^{2} - 1104 T^{3} + 10991 T^{4} - 68184 T^{5} + 502777 T^{6} - 2583456 T^{7} + 502777 p T^{8} - 68184 p^{2} T^{9} + 10991 p^{3} T^{10} - 1104 p^{4} T^{11} + 151 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 18 T + 331 T^{2} - 3726 T^{3} + 40271 T^{4} - 329286 T^{5} + 2565505 T^{6} - 15995132 T^{7} + 2565505 p T^{8} - 329286 p^{2} T^{9} + 40271 p^{3} T^{10} - 3726 p^{4} T^{11} + 331 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 3 T + 115 T^{2} + 46 T^{3} + 6937 T^{4} - 2243 T^{5} + 400035 T^{6} + 71428 T^{7} + 400035 p T^{8} - 2243 p^{2} T^{9} + 6937 p^{3} T^{10} + 46 p^{4} T^{11} + 115 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 11 T + 155 T^{2} - 802 T^{3} + 5659 T^{4} + 7401 T^{5} - 66167 T^{6} + 1842992 T^{7} - 66167 p T^{8} + 7401 p^{2} T^{9} + 5659 p^{3} T^{10} - 802 p^{4} T^{11} + 155 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 18 T + 261 T^{2} - 52 p T^{3} + 19817 T^{4} - 125422 T^{5} + 791045 T^{6} - 4841064 T^{7} + 791045 p T^{8} - 125422 p^{2} T^{9} + 19817 p^{3} T^{10} - 52 p^{5} T^{11} + 261 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 9 T + 235 T^{2} - 1546 T^{3} + 24925 T^{4} - 133939 T^{5} + 1738575 T^{6} - 8098276 T^{7} + 1738575 p T^{8} - 133939 p^{2} T^{9} + 24925 p^{3} T^{10} - 1546 p^{4} T^{11} + 235 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 6 T + 285 T^{2} + 1976 T^{3} + 38649 T^{4} + 277074 T^{5} + 3311861 T^{6} + 21400512 T^{7} + 3311861 p T^{8} + 277074 p^{2} T^{9} + 38649 p^{3} T^{10} + 1976 p^{4} T^{11} + 285 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 31 T + 535 T^{2} - 6790 T^{3} + 69225 T^{4} - 591481 T^{5} + 4616327 T^{6} - 35610100 T^{7} + 4616327 p T^{8} - 591481 p^{2} T^{9} + 69225 p^{3} T^{10} - 6790 p^{4} T^{11} + 535 p^{5} T^{12} - 31 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 121 T^{2} - 88 T^{3} + 11961 T^{4} + 740 T^{5} + 1136889 T^{6} - 1922896 T^{7} + 1136889 p T^{8} + 740 p^{2} T^{9} + 11961 p^{3} T^{10} - 88 p^{4} T^{11} + 121 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 3 T + 341 T^{2} - 978 T^{3} + 57149 T^{4} - 144615 T^{5} + 6018489 T^{6} - 12799064 T^{7} + 6018489 p T^{8} - 144615 p^{2} T^{9} + 57149 p^{3} T^{10} - 978 p^{4} T^{11} + 341 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 16 T + 387 T^{2} - 3552 T^{3} + 51721 T^{4} - 326192 T^{5} + 4263651 T^{6} - 23161280 T^{7} + 4263651 p T^{8} - 326192 p^{2} T^{9} + 51721 p^{3} T^{10} - 3552 p^{4} T^{11} + 387 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 34 T + 873 T^{2} - 15748 T^{3} + 239749 T^{4} - 2995550 T^{5} + 32955581 T^{6} - 310269816 T^{7} + 32955581 p T^{8} - 2995550 p^{2} T^{9} + 239749 p^{3} T^{10} - 15748 p^{4} T^{11} + 873 p^{5} T^{12} - 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 10 T + 277 T^{2} + 2852 T^{3} + 43165 T^{4} + 439462 T^{5} + 4766961 T^{6} + 43322104 T^{7} + 4766961 p T^{8} + 439462 p^{2} T^{9} + 43165 p^{3} T^{10} + 2852 p^{4} T^{11} + 277 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 24 T + 471 T^{2} + 5474 T^{3} + 68147 T^{4} + 725800 T^{5} + 9056177 T^{6} + 86973204 T^{7} + 9056177 p T^{8} + 725800 p^{2} T^{9} + 68147 p^{3} T^{10} + 5474 p^{4} T^{11} + 471 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 27 T + 663 T^{2} - 10270 T^{3} + 157605 T^{4} - 1896537 T^{5} + 23096995 T^{6} - 228012012 T^{7} + 23096995 p T^{8} - 1896537 p^{2} T^{9} + 157605 p^{3} T^{10} - 10270 p^{4} T^{11} + 663 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.85519823064285528412602563711, −3.80331145926147481885727764936, −3.76786814137381114876025233296, −3.71680977774950329439366530380, −3.40602594595561144895932914167, −3.26408040428922637684516773302, −3.00849033038012707074341871339, −3.00282293180452128939784060336, −2.94892764203252726203754431168, −2.93897774400695017140001487254, −2.82674034228013454451627780037, −2.37589968161375351125482428354, −2.27133506427744724705295000907, −2.22873203053594343150436770928, −2.09058436473679784995556583134, −1.97779259543198923467961324664, −1.84772986138652466546077392090, −1.81918499645150590272642314067, −1.35484446575397674918582154273, −1.28821179269948644237831527136, −1.09471953545226389335134056670, −0.895087563785229749375475346493, −0.841068026888373818739008716169, −0.61584901433793731617846887997, −0.53920289515003282918328962168, 0.53920289515003282918328962168, 0.61584901433793731617846887997, 0.841068026888373818739008716169, 0.895087563785229749375475346493, 1.09471953545226389335134056670, 1.28821179269948644237831527136, 1.35484446575397674918582154273, 1.81918499645150590272642314067, 1.84772986138652466546077392090, 1.97779259543198923467961324664, 2.09058436473679784995556583134, 2.22873203053594343150436770928, 2.27133506427744724705295000907, 2.37589968161375351125482428354, 2.82674034228013454451627780037, 2.93897774400695017140001487254, 2.94892764203252726203754431168, 3.00282293180452128939784060336, 3.00849033038012707074341871339, 3.26408040428922637684516773302, 3.40602594595561144895932914167, 3.71680977774950329439366530380, 3.76786814137381114876025233296, 3.80331145926147481885727764936, 3.85519823064285528412602563711
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.