Dirichlet series
| L(s) = 1 | + 8·2-s + 36·4-s − 5·7-s + 120·8-s − 6·11-s − 8·13-s − 40·14-s + 330·16-s + 5·17-s − 14·19-s − 48·22-s − 21·23-s − 23·25-s − 64·26-s − 180·28-s − 3·29-s − 4·31-s + 792·32-s + 40·34-s − 3·37-s − 112·38-s − 7·41-s − 12·43-s − 216·44-s − 168·46-s − 25·47-s − 19·49-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s − 1.88·7-s + 42.4·8-s − 1.80·11-s − 2.21·13-s − 10.6·14-s + 82.5·16-s + 1.21·17-s − 3.21·19-s − 10.2·22-s − 4.37·23-s − 4.59·25-s − 12.5·26-s − 34.0·28-s − 0.557·29-s − 0.718·31-s + 140.·32-s + 6.85·34-s − 0.493·37-s − 18.1·38-s − 1.09·41-s − 1.82·43-s − 32.5·44-s − 24.7·46-s − 3.64·47-s − 2.71·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 149^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 149^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{24} \cdot 149^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.90306\times 10^{14}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{24} \cdot 149^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 \) | |
| 149 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + 23 T^{2} + 2 T^{3} + 261 T^{4} + 68 T^{5} + 1964 T^{6} + 734 T^{7} + 11099 T^{8} + 734 p T^{9} + 1964 p^{2} T^{10} + 68 p^{3} T^{11} + 261 p^{4} T^{12} + 2 p^{5} T^{13} + 23 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 5 T + 44 T^{2} + 164 T^{3} + 122 p T^{4} + 2581 T^{5} + 208 p^{2} T^{6} + 25906 T^{7} + 84279 T^{8} + 25906 p T^{9} + 208 p^{4} T^{10} + 2581 p^{3} T^{11} + 122 p^{5} T^{12} + 164 p^{5} T^{13} + 44 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 6 T + 69 T^{2} + 320 T^{3} + 2012 T^{4} + 7675 T^{5} + 34896 T^{6} + 115513 T^{7} + 435155 T^{8} + 115513 p T^{9} + 34896 p^{2} T^{10} + 7675 p^{3} T^{11} + 2012 p^{4} T^{12} + 320 p^{5} T^{13} + 69 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 8 T + 84 T^{2} + 448 T^{3} + 2738 T^{4} + 10836 T^{5} + 50511 T^{6} + 166032 T^{7} + 700113 T^{8} + 166032 p T^{9} + 50511 p^{2} T^{10} + 10836 p^{3} T^{11} + 2738 p^{4} T^{12} + 448 p^{5} T^{13} + 84 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 5 T + 98 T^{2} - 386 T^{3} + 4323 T^{4} - 14035 T^{5} + 119347 T^{6} - 330216 T^{7} + 2354845 T^{8} - 330216 p T^{9} + 119347 p^{2} T^{10} - 14035 p^{3} T^{11} + 4323 p^{4} T^{12} - 386 p^{5} T^{13} + 98 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 14 T + 189 T^{2} + 1675 T^{3} + 13551 T^{4} + 88651 T^{5} + 528103 T^{6} + 2701748 T^{7} + 12614699 T^{8} + 2701748 p T^{9} + 528103 p^{2} T^{10} + 88651 p^{3} T^{11} + 13551 p^{4} T^{12} + 1675 p^{5} T^{13} + 189 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 21 T + 315 T^{2} + 3351 T^{3} + 29840 T^{4} + 220880 T^{5} + 1441926 T^{6} + 8201727 T^{7} + 41898423 T^{8} + 8201727 p T^{9} + 1441926 p^{2} T^{10} + 220880 p^{3} T^{11} + 29840 p^{4} T^{12} + 3351 p^{5} T^{13} + 315 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 3 T + 4 p T^{2} + 257 T^{3} + 5913 T^{4} + 5002 T^{5} + 188655 T^{6} - 144119 T^{7} + 5241187 T^{8} - 144119 p T^{9} + 188655 p^{2} T^{10} + 5002 p^{3} T^{11} + 5913 p^{4} T^{12} + 257 p^{5} T^{13} + 4 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 4 T + 143 T^{2} + 277 T^{3} + 9440 T^{4} + 5823 T^{5} + 413774 T^{6} - 77032 T^{7} + 14007581 T^{8} - 77032 p T^{9} + 413774 p^{2} T^{10} + 5823 p^{3} T^{11} + 9440 p^{4} T^{12} + 277 p^{5} T^{13} + 143 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 3 T + 157 T^{2} + 169 T^{3} + 10925 T^{4} - 4578 T^{5} + 520980 T^{6} - 602386 T^{7} + 20795125 T^{8} - 602386 p T^{9} + 520980 p^{2} T^{10} - 4578 p^{3} T^{11} + 10925 p^{4} T^{12} + 169 p^{5} T^{13} + 157 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 7 T + 221 T^{2} + 1651 T^{3} + 22651 T^{4} + 175892 T^{5} + 1468078 T^{6} + 11131140 T^{7} + 69043319 T^{8} + 11131140 p T^{9} + 1468078 p^{2} T^{10} + 175892 p^{3} T^{11} + 22651 p^{4} T^{12} + 1651 p^{5} T^{13} + 221 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 12 T + 268 T^{2} + 2841 T^{3} + 34765 T^{4} + 312940 T^{5} + 2760526 T^{6} + 20722959 T^{7} + 144745419 T^{8} + 20722959 p T^{9} + 2760526 p^{2} T^{10} + 312940 p^{3} T^{11} + 34765 p^{4} T^{12} + 2841 p^{5} T^{13} + 268 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 25 T + 534 T^{2} + 7809 T^{3} + 2117 p T^{4} + 1049944 T^{5} + 9811105 T^{6} + 80167517 T^{7} + 583909437 T^{8} + 80167517 p T^{9} + 9811105 p^{2} T^{10} + 1049944 p^{3} T^{11} + 2117 p^{5} T^{12} + 7809 p^{5} T^{13} + 534 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 3 T + 259 T^{2} + 671 T^{3} + 33140 T^{4} + 76472 T^{5} + 2800566 T^{6} + 110221 p T^{7} + 172037347 T^{8} + 110221 p^{2} T^{9} + 2800566 p^{2} T^{10} + 76472 p^{3} T^{11} + 33140 p^{4} T^{12} + 671 p^{5} T^{13} + 259 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 2 T + 187 T^{2} + 781 T^{3} + 15293 T^{4} + 93219 T^{5} + 839319 T^{6} + 6407678 T^{7} + 44695837 T^{8} + 6407678 p T^{9} + 839319 p^{2} T^{10} + 93219 p^{3} T^{11} + 15293 p^{4} T^{12} + 781 p^{5} T^{13} + 187 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 17 T + 287 T^{2} + 3211 T^{3} + 40377 T^{4} + 388503 T^{5} + 3878930 T^{6} + 31804176 T^{7} + 272510093 T^{8} + 31804176 p T^{9} + 3878930 p^{2} T^{10} + 388503 p^{3} T^{11} + 40377 p^{4} T^{12} + 3211 p^{5} T^{13} + 287 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 14 T + 325 T^{2} + 3419 T^{3} + 55257 T^{4} + 492865 T^{5} + 6126075 T^{6} + 46584916 T^{7} + 484079105 T^{8} + 46584916 p T^{9} + 6126075 p^{2} T^{10} + 492865 p^{3} T^{11} + 55257 p^{4} T^{12} + 3419 p^{5} T^{13} + 325 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 7 T + 376 T^{2} + 2079 T^{3} + 67454 T^{4} + 298691 T^{5} + 7720851 T^{6} + 28228478 T^{7} + 635043539 T^{8} + 28228478 p T^{9} + 7720851 p^{2} T^{10} + 298691 p^{3} T^{11} + 67454 p^{4} T^{12} + 2079 p^{5} T^{13} + 376 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 10 T + 380 T^{2} + 3079 T^{3} + 71273 T^{4} + 493354 T^{5} + 8654984 T^{6} + 51366911 T^{7} + 741758925 T^{8} + 51366911 p T^{9} + 8654984 p^{2} T^{10} + 493354 p^{3} T^{11} + 71273 p^{4} T^{12} + 3079 p^{5} T^{13} + 380 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 33 T + 891 T^{2} + 16593 T^{3} + 267540 T^{4} + 3546960 T^{5} + 42232098 T^{6} + 436784437 T^{7} + 4134028575 T^{8} + 436784437 p T^{9} + 42232098 p^{2} T^{10} + 3546960 p^{3} T^{11} + 267540 p^{4} T^{12} + 16593 p^{5} T^{13} + 891 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 13 T + 566 T^{2} + 6044 T^{3} + 145132 T^{4} + 1282046 T^{5} + 22184174 T^{6} + 163089004 T^{7} + 2235297087 T^{8} + 163089004 p T^{9} + 22184174 p^{2} T^{10} + 1282046 p^{3} T^{11} + 145132 p^{4} T^{12} + 6044 p^{5} T^{13} + 566 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 22 T + 483 T^{2} - 6152 T^{3} + 66677 T^{4} - 418457 T^{5} + 1196132 T^{6} + 23297564 T^{7} - 288951823 T^{8} + 23297564 p T^{9} + 1196132 p^{2} T^{10} - 418457 p^{3} T^{11} + 66677 p^{4} T^{12} - 6152 p^{5} T^{13} + 483 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 11 T + 555 T^{2} + 4180 T^{3} + 124875 T^{4} + 596251 T^{5} + 16127779 T^{6} + 47076750 T^{7} + 1614911481 T^{8} + 47076750 p T^{9} + 16127779 p^{2} T^{10} + 596251 p^{3} T^{11} + 124875 p^{4} T^{12} + 4180 p^{5} T^{13} + 555 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.72935469909954844465906017686, −3.46279417612143003195773554899, −3.39750194153659680168080021874, −3.38938134322790743823689423077, −3.17210789899354497059426356554, −3.11928158919795793937981071835, −3.07897307867997208437588388750, −2.95679218742173203016473706123, −2.87729104964500984998550256708, −2.68167142082102771160215082940, −2.46495984259136681361221161002, −2.44393772393431251360062665533, −2.43813670173509841599752120866, −2.35781902147189453584049878193, −2.33315891407980221618081312271, −2.17445762736279473930722791843, −2.10456334542837513793378179430, −1.72675422829293303923482752340, −1.60299192201221603361601808228, −1.57477545734102028870811201116, −1.55434419022248701766580676549, −1.55374227226669598553487297063, −1.47933780090759696100823868337, −1.32770669344170694071414737484, −1.12277599167935978311161255993, 0, 0, 0, 0, 0, 0, 0, 0, 1.12277599167935978311161255993, 1.32770669344170694071414737484, 1.47933780090759696100823868337, 1.55374227226669598553487297063, 1.55434419022248701766580676549, 1.57477545734102028870811201116, 1.60299192201221603361601808228, 1.72675422829293303923482752340, 2.10456334542837513793378179430, 2.17445762736279473930722791843, 2.33315891407980221618081312271, 2.35781902147189453584049878193, 2.43813670173509841599752120866, 2.44393772393431251360062665533, 2.46495984259136681361221161002, 2.68167142082102771160215082940, 2.87729104964500984998550256708, 2.95679218742173203016473706123, 3.07897307867997208437588388750, 3.11928158919795793937981071835, 3.17210789899354497059426356554, 3.38938134322790743823689423077, 3.39750194153659680168080021874, 3.46279417612143003195773554899, 3.72935469909954844465906017686
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.