Dirichlet series
| L(s) = 1 | − 2·2-s + 8·3-s − 3·4-s − 2·5-s − 16·6-s − 8·7-s + 6·8-s + 36·9-s + 4·10-s − 8·11-s − 24·12-s + 16·14-s − 16·15-s + 8·16-s + 10·17-s − 72·18-s − 18·19-s + 6·20-s − 64·21-s + 16·22-s − 2·23-s + 48·24-s − 21·25-s + 120·27-s + 24·28-s − 12·29-s + 32·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4.61·3-s − 3/2·4-s − 0.894·5-s − 6.53·6-s − 3.02·7-s + 2.12·8-s + 12·9-s + 1.26·10-s − 2.41·11-s − 6.92·12-s + 4.27·14-s − 4.13·15-s + 2·16-s + 2.42·17-s − 16.9·18-s − 4.12·19-s + 1.34·20-s − 13.9·21-s + 3.41·22-s − 0.417·23-s + 9.79·24-s − 4.19·25-s + 23.0·27-s + 4.53·28-s − 2.22·29-s + 5.84·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 7^{8} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.15972\times 10^{11}\) |
| Root analytic conductor: | \(5.32343\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{8} \cdot 7^{8} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 3 | \( ( 1 - T )^{8} \) |
| 7 | \( ( 1 + T )^{8} \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T + 7 T^{2} + 7 p T^{3} + 29 T^{4} + 13 p^{2} T^{5} + 23 p^{2} T^{6} + 17 p^{3} T^{7} + 217 T^{8} + 17 p^{4} T^{9} + 23 p^{4} T^{10} + 13 p^{5} T^{11} + 29 p^{4} T^{12} + 7 p^{6} T^{13} + 7 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 2 T + p^{2} T^{2} + 46 T^{3} + 309 T^{4} + 508 T^{5} + 2499 T^{6} + 3572 T^{7} + 14537 T^{8} + 3572 p T^{9} + 2499 p^{2} T^{10} + 508 p^{3} T^{11} + 309 p^{4} T^{12} + 46 p^{5} T^{13} + p^{8} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 8 T + 76 T^{2} + 478 T^{3} + 2697 T^{4} + 13138 T^{5} + 57165 T^{6} + 218756 T^{7} + 777974 T^{8} + 218756 p T^{9} + 57165 p^{2} T^{10} + 13138 p^{3} T^{11} + 2697 p^{4} T^{12} + 478 p^{5} T^{13} + 76 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 10 T + 81 T^{2} - 392 T^{3} + 2039 T^{4} - 8668 T^{5} + 49629 T^{6} - 223908 T^{7} + 1084009 T^{8} - 223908 p T^{9} + 49629 p^{2} T^{10} - 8668 p^{3} T^{11} + 2039 p^{4} T^{12} - 392 p^{5} T^{13} + 81 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 18 T + 219 T^{2} + 1802 T^{3} + 12493 T^{4} + 72104 T^{5} + 389638 T^{6} + 1885648 T^{7} + 8689070 T^{8} + 1885648 p T^{9} + 389638 p^{2} T^{10} + 72104 p^{3} T^{11} + 12493 p^{4} T^{12} + 1802 p^{5} T^{13} + 219 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 87 T^{2} + 320 T^{3} + 4284 T^{4} + 19070 T^{5} + 146950 T^{6} + 679968 T^{7} + 3822636 T^{8} + 679968 p T^{9} + 146950 p^{2} T^{10} + 19070 p^{3} T^{11} + 4284 p^{4} T^{12} + 320 p^{5} T^{13} + 87 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 12 T + 6 p T^{2} + 1316 T^{3} + 11829 T^{4} + 71260 T^{5} + 524191 T^{6} + 2765850 T^{7} + 17529123 T^{8} + 2765850 p T^{9} + 524191 p^{2} T^{10} + 71260 p^{3} T^{11} + 11829 p^{4} T^{12} + 1316 p^{5} T^{13} + 6 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 16 T + 265 T^{2} + 2662 T^{3} + 25538 T^{4} + 190462 T^{5} + 1354956 T^{6} + 8268280 T^{7} + 48862608 T^{8} + 8268280 p T^{9} + 1354956 p^{2} T^{10} + 190462 p^{3} T^{11} + 25538 p^{4} T^{12} + 2662 p^{5} T^{13} + 265 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 24 T + 415 T^{2} + 5212 T^{3} + 55881 T^{4} + 506640 T^{5} + 4102323 T^{6} + 29242236 T^{7} + 188735749 T^{8} + 29242236 p T^{9} + 4102323 p^{2} T^{10} + 506640 p^{3} T^{11} + 55881 p^{4} T^{12} + 5212 p^{5} T^{13} + 415 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4 T + 245 T^{2} - 1028 T^{3} + 28710 T^{4} - 115756 T^{5} + 51139 p T^{6} - 7523404 T^{7} + 103526898 T^{8} - 7523404 p T^{9} + 51139 p^{3} T^{10} - 115756 p^{3} T^{11} + 28710 p^{4} T^{12} - 1028 p^{5} T^{13} + 245 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 10 T + 229 T^{2} + 2092 T^{3} + 27208 T^{4} + 212444 T^{5} + 2061456 T^{6} + 13525358 T^{7} + 106372192 T^{8} + 13525358 p T^{9} + 2061456 p^{2} T^{10} + 212444 p^{3} T^{11} + 27208 p^{4} T^{12} + 2092 p^{5} T^{13} + 229 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 10 T + 315 T^{2} + 2812 T^{3} + 45700 T^{4} + 356574 T^{5} + 3982814 T^{6} + 26481924 T^{7} + 228132200 T^{8} + 26481924 p T^{9} + 3982814 p^{2} T^{10} + 356574 p^{3} T^{11} + 45700 p^{4} T^{12} + 2812 p^{5} T^{13} + 315 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 256 T^{2} - 1200 T^{3} + 27982 T^{4} - 101214 T^{5} + 1836448 T^{6} - 5401950 T^{7} + 97827967 T^{8} - 5401950 p T^{9} + 1836448 p^{2} T^{10} - 101214 p^{3} T^{11} + 27982 p^{4} T^{12} - 1200 p^{5} T^{13} + 256 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 6 T + 365 T^{2} + 1958 T^{3} + 61846 T^{4} + 296850 T^{5} + 6442776 T^{6} + 27026362 T^{7} + 455305316 T^{8} + 27026362 p T^{9} + 6442776 p^{2} T^{10} + 296850 p^{3} T^{11} + 61846 p^{4} T^{12} + 1958 p^{5} T^{13} + 365 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 6 T + 256 T^{2} - 1160 T^{3} + 28470 T^{4} - 97782 T^{5} + 1977216 T^{6} - 5451270 T^{7} + 117546919 T^{8} - 5451270 p T^{9} + 1977216 p^{2} T^{10} - 97782 p^{3} T^{11} + 28470 p^{4} T^{12} - 1160 p^{5} T^{13} + 256 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 24 T + 605 T^{2} + 9548 T^{3} + 144468 T^{4} + 1717876 T^{5} + 19218620 T^{6} + 181264624 T^{7} + 1601805772 T^{8} + 181264624 p T^{9} + 19218620 p^{2} T^{10} + 1717876 p^{3} T^{11} + 144468 p^{4} T^{12} + 9548 p^{5} T^{13} + 605 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 42 T + 1004 T^{2} + 16258 T^{3} + 205691 T^{4} + 2191008 T^{5} + 21477229 T^{6} + 198381736 T^{7} + 1735761210 T^{8} + 198381736 p T^{9} + 21477229 p^{2} T^{10} + 2191008 p^{3} T^{11} + 205691 p^{4} T^{12} + 16258 p^{5} T^{13} + 1004 p^{6} T^{14} + 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 32 T + 773 T^{2} + 13022 T^{3} + 189699 T^{4} + 2295940 T^{5} + 25261697 T^{6} + 244556986 T^{7} + 2206859169 T^{8} + 244556986 p T^{9} + 25261697 p^{2} T^{10} + 2295940 p^{3} T^{11} + 189699 p^{4} T^{12} + 13022 p^{5} T^{13} + 773 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 2 T + 69 T^{2} + 132 T^{3} + 7640 T^{4} - 51688 T^{5} + 352216 T^{6} - 4388422 T^{7} + 25667720 T^{8} - 4388422 p T^{9} + 352216 p^{2} T^{10} - 51688 p^{3} T^{11} + 7640 p^{4} T^{12} + 132 p^{5} T^{13} + 69 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 2 T + 324 T^{2} + 248 T^{3} + 45391 T^{4} + 224408 T^{5} + 3677525 T^{6} + 40037950 T^{7} + 262136458 T^{8} + 40037950 p T^{9} + 3677525 p^{2} T^{10} + 224408 p^{3} T^{11} + 45391 p^{4} T^{12} + 248 p^{5} T^{13} + 324 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 12 T + 119 T^{2} - 424 T^{3} + 8033 T^{4} + 114408 T^{5} + 2438338 T^{6} + 4283468 T^{7} + 79370814 T^{8} + 4283468 p T^{9} + 2438338 p^{2} T^{10} + 114408 p^{3} T^{11} + 8033 p^{4} T^{12} - 424 p^{5} T^{13} + 119 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 64 T + 2281 T^{2} + 58078 T^{3} + 1164026 T^{4} + 19256938 T^{5} + 270459048 T^{6} + 3276822736 T^{7} + 34549999164 T^{8} + 3276822736 p T^{9} + 270459048 p^{2} T^{10} + 19256938 p^{3} T^{11} + 1164026 p^{4} T^{12} + 58078 p^{5} T^{13} + 2281 p^{6} T^{14} + 64 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.03667882004356461088757071440, −3.67763923382302315737439394657, −3.66521784509302450539583517295, −3.63828537232105531652465196026, −3.58531306912470726730271269726, −3.39749926116390900514272247598, −3.31663510059474042165860961640, −3.10671092987798521382663998396, −3.03957157553691404000900357434, −3.03354816953053997633211933098, −2.84377383581018158888410372273, −2.77851925094013137466007471259, −2.64448816550699213799057172250, −2.56277161675716100053017917783, −2.49514950227969214407742663473, −2.22058275560937481563322001178, −2.11467853149638461841622024484, −1.91452614718409322420414540466, −1.85163560234550183868632260318, −1.77989280716960204937371048364, −1.57122811404988768961709164080, −1.46242180620499899088928254600, −1.32168063045229017690289531950, −1.23823868664664815971395117683, −1.09914074459243964220134502034, 0, 0, 0, 0, 0, 0, 0, 0, 1.09914074459243964220134502034, 1.23823868664664815971395117683, 1.32168063045229017690289531950, 1.46242180620499899088928254600, 1.57122811404988768961709164080, 1.77989280716960204937371048364, 1.85163560234550183868632260318, 1.91452614718409322420414540466, 2.11467853149638461841622024484, 2.22058275560937481563322001178, 2.49514950227969214407742663473, 2.56277161675716100053017917783, 2.64448816550699213799057172250, 2.77851925094013137466007471259, 2.84377383581018158888410372273, 3.03354816953053997633211933098, 3.03957157553691404000900357434, 3.10671092987798521382663998396, 3.31663510059474042165860961640, 3.39749926116390900514272247598, 3.58531306912470726730271269726, 3.63828537232105531652465196026, 3.66521784509302450539583517295, 3.67763923382302315737439394657, 4.03667882004356461088757071440
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.