Dirichlet series
| L(s) = 1 | + 4·2-s + 3·3-s + 2·4-s + 12·6-s + 7·7-s − 14·8-s − 7·9-s − 18·11-s + 6·12-s + 13-s + 28·14-s − 22·16-s + 2·17-s − 28·18-s − 6·19-s + 21·21-s − 72·22-s + 22·23-s − 42·24-s + 4·26-s − 34·27-s + 14·28-s − 16·29-s − 18·31-s + 14·32-s − 54·33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.73·3-s + 4-s + 4.89·6-s + 2.64·7-s − 4.94·8-s − 7/3·9-s − 5.42·11-s + 1.73·12-s + 0.277·13-s + 7.48·14-s − 5.5·16-s + 0.485·17-s − 6.59·18-s − 1.37·19-s + 4.58·21-s − 15.3·22-s + 4.58·23-s − 8.57·24-s + 0.784·26-s − 6.54·27-s + 2.64·28-s − 2.97·29-s − 3.23·31-s + 2.47·32-s − 9.40·33-s + 1.37·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{14} \cdot 241^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{14} \cdot 241^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(5.96539\times 10^{11}\) |
| Root analytic conductor: | \(6.93612\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 5^{14} \cdot 241^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 241 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p^{2} T + 7 p T^{2} - 17 p T^{3} + 37 p T^{4} - 67 p T^{5} + 223 T^{6} - 327 T^{7} + 223 p T^{8} - 67 p^{3} T^{9} + 37 p^{4} T^{10} - 17 p^{5} T^{11} + 7 p^{6} T^{12} - p^{8} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - p T + 16 T^{2} - 35 T^{3} + 110 T^{4} - 191 T^{5} + 467 T^{6} - 679 T^{7} + 467 p T^{8} - 191 p^{2} T^{9} + 110 p^{3} T^{10} - 35 p^{4} T^{11} + 16 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - p T + 46 T^{2} - 4 p^{2} T^{3} + 786 T^{4} - 2528 T^{5} + 7897 T^{6} - 21047 T^{7} + 7897 p T^{8} - 2528 p^{2} T^{9} + 786 p^{3} T^{10} - 4 p^{6} T^{11} + 46 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 18 T + 194 T^{2} + 1471 T^{3} + 8839 T^{4} + 43563 T^{5} + 182353 T^{6} + 651389 T^{7} + 182353 p T^{8} + 43563 p^{2} T^{9} + 8839 p^{3} T^{10} + 1471 p^{4} T^{11} + 194 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - T + 43 T^{2} - 16 T^{3} + 74 p T^{4} - 171 T^{5} + 1275 p T^{6} - 3431 T^{7} + 1275 p^{2} T^{8} - 171 p^{2} T^{9} + 74 p^{4} T^{10} - 16 p^{4} T^{11} + 43 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 2 T + 54 T^{2} - 118 T^{3} + 1511 T^{4} - 4204 T^{5} + 32789 T^{6} - 93345 T^{7} + 32789 p T^{8} - 4204 p^{2} T^{9} + 1511 p^{3} T^{10} - 118 p^{4} T^{11} + 54 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 6 T + 77 T^{2} + 408 T^{3} + 3328 T^{4} + 14851 T^{5} + 91875 T^{6} + 346087 T^{7} + 91875 p T^{8} + 14851 p^{2} T^{9} + 3328 p^{3} T^{10} + 408 p^{4} T^{11} + 77 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 22 T + 329 T^{2} - 3499 T^{3} + 30354 T^{4} - 215473 T^{5} + 1308858 T^{6} - 6746533 T^{7} + 1308858 p T^{8} - 215473 p^{2} T^{9} + 30354 p^{3} T^{10} - 3499 p^{4} T^{11} + 329 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 16 T + 228 T^{2} + 1945 T^{3} + 15113 T^{4} + 86708 T^{5} + 503802 T^{6} + 2527253 T^{7} + 503802 p T^{8} + 86708 p^{2} T^{9} + 15113 p^{3} T^{10} + 1945 p^{4} T^{11} + 228 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 18 T + 321 T^{2} + 3457 T^{3} + 35295 T^{4} + 269386 T^{5} + 1944797 T^{6} + 11129437 T^{7} + 1944797 p T^{8} + 269386 p^{2} T^{9} + 35295 p^{3} T^{10} + 3457 p^{4} T^{11} + 321 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 8 T + 140 T^{2} + 446 T^{3} + 5906 T^{4} - 5317 T^{5} + 140628 T^{6} - 725991 T^{7} + 140628 p T^{8} - 5317 p^{2} T^{9} + 5906 p^{3} T^{10} + 446 p^{4} T^{11} + 140 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 15 T + 165 T^{2} + 1716 T^{3} + 15349 T^{4} + 112837 T^{5} + 825907 T^{6} + 5652081 T^{7} + 825907 p T^{8} + 112837 p^{2} T^{9} + 15349 p^{3} T^{10} + 1716 p^{4} T^{11} + 165 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 14 T + 160 T^{2} + 1152 T^{3} + 8465 T^{4} + 39218 T^{5} + 236797 T^{6} + 1042279 T^{7} + 236797 p T^{8} + 39218 p^{2} T^{9} + 8465 p^{3} T^{10} + 1152 p^{4} T^{11} + 160 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 10 T + 213 T^{2} - 1823 T^{3} + 23033 T^{4} - 162514 T^{5} + 1573229 T^{6} - 9290969 T^{7} + 1573229 p T^{8} - 162514 p^{2} T^{9} + 23033 p^{3} T^{10} - 1823 p^{4} T^{11} + 213 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 15 T + 248 T^{2} + 2459 T^{3} + 26359 T^{4} + 226500 T^{5} + 2031262 T^{6} + 14891311 T^{7} + 2031262 p T^{8} + 226500 p^{2} T^{9} + 26359 p^{3} T^{10} + 2459 p^{4} T^{11} + 248 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 18 T + 283 T^{2} + 3342 T^{3} + 39787 T^{4} + 377400 T^{5} + 3440044 T^{6} + 26583077 T^{7} + 3440044 p T^{8} + 377400 p^{2} T^{9} + 39787 p^{3} T^{10} + 3342 p^{4} T^{11} + 283 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 173 T^{2} + 229 T^{3} + 13815 T^{4} + 53354 T^{5} + 1189845 T^{6} + 3012271 T^{7} + 1189845 p T^{8} + 53354 p^{2} T^{9} + 13815 p^{3} T^{10} + 229 p^{4} T^{11} + 173 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 18 T + 312 T^{2} + 3657 T^{3} + 46177 T^{4} + 442394 T^{5} + 4388546 T^{6} + 34987571 T^{7} + 4388546 p T^{8} + 442394 p^{2} T^{9} + 46177 p^{3} T^{10} + 3657 p^{4} T^{11} + 312 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 50 T + 1452 T^{2} + 29886 T^{3} + 479876 T^{4} + 6256313 T^{5} + 67998420 T^{6} + 622620957 T^{7} + 67998420 p T^{8} + 6256313 p^{2} T^{9} + 479876 p^{3} T^{10} + 29886 p^{4} T^{11} + 1452 p^{5} T^{12} + 50 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 133 T^{2} - 1068 T^{3} + 10948 T^{4} - 119175 T^{5} + 1571649 T^{6} - 6028685 T^{7} + 1571649 p T^{8} - 119175 p^{2} T^{9} + 10948 p^{3} T^{10} - 1068 p^{4} T^{11} + 133 p^{5} T^{12} + p^{7} T^{14} \) | |
| 79 | \( 1 + 15 T + 468 T^{2} + 5553 T^{3} + 1243 p T^{4} + 946068 T^{5} + 12199714 T^{6} + 95010077 T^{7} + 12199714 p T^{8} + 946068 p^{2} T^{9} + 1243 p^{4} T^{10} + 5553 p^{4} T^{11} + 468 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 24 T + 705 T^{2} - 11938 T^{3} + 194891 T^{4} - 2473798 T^{5} + 28249166 T^{6} - 273618813 T^{7} + 28249166 p T^{8} - 2473798 p^{2} T^{9} + 194891 p^{3} T^{10} - 11938 p^{4} T^{11} + 705 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 13 T + 260 T^{2} + 2275 T^{3} + 16962 T^{4} + 2186 T^{5} - 807494 T^{6} - 17411725 T^{7} - 807494 p T^{8} + 2186 p^{2} T^{9} + 16962 p^{3} T^{10} + 2275 p^{4} T^{11} + 260 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + T + 396 T^{2} - 357 T^{3} + 75395 T^{4} - 180194 T^{5} + 9562600 T^{6} - 26454385 T^{7} + 9562600 p T^{8} - 180194 p^{2} T^{9} + 75395 p^{3} T^{10} - 357 p^{4} T^{11} + 396 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.00124615383734655925149610316, −3.91690973256586203223628602336, −3.81981217333182225157460398968, −3.71447690082745378758782359501, −3.42874127235488098654666869704, −3.35358439270987767988950283187, −3.32237164667051283804748097100, −3.27782049858611290373735521444, −3.26537205715113307767772010714, −3.09237554360249367464378793240, −3.05791456243855914230237755041, −2.77917796061170350019401807369, −2.63420767720936017327235550520, −2.61751433076654176149279470924, −2.58066776394611202272409581501, −2.31185049944708379447400409459, −2.28821761606705352581899603590, −2.00082004079458772946180460546, −1.87012585891195728389189066642, −1.83045457401636296032668057343, −1.51514474603164630693546597119, −1.46950034316185088932094245061, −1.38475331285569718575612177361, −1.05555259906198372615003846735, −1.04486726593146973910804971440, 0, 0, 0, 0, 0, 0, 0, 1.04486726593146973910804971440, 1.05555259906198372615003846735, 1.38475331285569718575612177361, 1.46950034316185088932094245061, 1.51514474603164630693546597119, 1.83045457401636296032668057343, 1.87012585891195728389189066642, 2.00082004079458772946180460546, 2.28821761606705352581899603590, 2.31185049944708379447400409459, 2.58066776394611202272409581501, 2.61751433076654176149279470924, 2.63420767720936017327235550520, 2.77917796061170350019401807369, 3.05791456243855914230237755041, 3.09237554360249367464378793240, 3.26537205715113307767772010714, 3.27782049858611290373735521444, 3.32237164667051283804748097100, 3.35358439270987767988950283187, 3.42874127235488098654666869704, 3.71447690082745378758782359501, 3.81981217333182225157460398968, 3.91690973256586203223628602336, 4.00124615383734655925149610316
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.