Dirichlet series
| L(s) = 1 | − 4·2-s + 3·4-s + 8·5-s − 8·7-s + 8·8-s − 32·10-s − 6·13-s + 32·14-s − 11·16-s − 8·17-s + 2·19-s + 24·20-s + 4·23-s + 36·25-s + 24·26-s − 24·28-s − 22·29-s + 10·31-s − 12·32-s + 32·34-s − 64·35-s − 14·37-s − 8·38-s + 64·40-s − 22·41-s − 14·43-s − 16·46-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3/2·4-s + 3.57·5-s − 3.02·7-s + 2.82·8-s − 10.1·10-s − 1.66·13-s + 8.55·14-s − 2.75·16-s − 1.94·17-s + 0.458·19-s + 5.36·20-s + 0.834·23-s + 36/5·25-s + 4.70·26-s − 4.53·28-s − 4.08·29-s + 1.79·31-s − 2.12·32-s + 5.48·34-s − 10.8·35-s − 2.30·37-s − 1.29·38-s + 10.1·40-s − 3.43·41-s − 2.13·43-s − 2.35·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.27702\times 10^{13}\) |
| Root analytic conductor: | \(6.59382\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( 1 \) |
| 5 | \( ( 1 - T )^{8} \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T + 13 T^{2} + p^{5} T^{3} + 17 p^{2} T^{4} + p^{7} T^{5} + 109 p T^{6} + 171 p T^{7} + 503 T^{8} + 171 p^{2} T^{9} + 109 p^{3} T^{10} + p^{10} T^{11} + 17 p^{6} T^{12} + p^{10} T^{13} + 13 p^{6} T^{14} + p^{9} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 8 T + 60 T^{2} + 306 T^{3} + 206 p T^{4} + 5512 T^{5} + 2801 p T^{6} + 59628 T^{7} + 169641 T^{8} + 59628 p T^{9} + 2801 p^{3} T^{10} + 5512 p^{3} T^{11} + 206 p^{5} T^{12} + 306 p^{5} T^{13} + 60 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 6 T + 67 T^{2} + 332 T^{3} + 2208 T^{4} + 9290 T^{5} + 47137 T^{6} + 169446 T^{7} + 715203 T^{8} + 169446 p T^{9} + 47137 p^{2} T^{10} + 9290 p^{3} T^{11} + 2208 p^{4} T^{12} + 332 p^{5} T^{13} + 67 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 8 T + 6 p T^{2} + 36 p T^{3} + 4776 T^{4} + 23500 T^{5} + 139644 T^{6} + 580348 T^{7} + 2826809 T^{8} + 580348 p T^{9} + 139644 p^{2} T^{10} + 23500 p^{3} T^{11} + 4776 p^{4} T^{12} + 36 p^{6} T^{13} + 6 p^{7} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 2 T + 61 T^{2} - 58 T^{3} + 1800 T^{4} - 1524 T^{5} + 44723 T^{6} - 67116 T^{7} + 981959 T^{8} - 67116 p T^{9} + 44723 p^{2} T^{10} - 1524 p^{3} T^{11} + 1800 p^{4} T^{12} - 58 p^{5} T^{13} + 61 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 4 T + 119 T^{2} - 26 p T^{3} + 7151 T^{4} - 36104 T^{5} + 12528 p T^{6} - 1249998 T^{7} + 8024929 T^{8} - 1249998 p T^{9} + 12528 p^{3} T^{10} - 36104 p^{3} T^{11} + 7151 p^{4} T^{12} - 26 p^{6} T^{13} + 119 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 22 T + 13 p T^{2} + 4560 T^{3} + 46794 T^{4} + 396994 T^{5} + 2964619 T^{6} + 19125834 T^{7} + 110217029 T^{8} + 19125834 p T^{9} + 2964619 p^{2} T^{10} + 396994 p^{3} T^{11} + 46794 p^{4} T^{12} + 4560 p^{5} T^{13} + 13 p^{7} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 10 T + 187 T^{2} - 1586 T^{3} + 17239 T^{4} - 119702 T^{5} + 973144 T^{6} - 5596480 T^{7} + 36519443 T^{8} - 5596480 p T^{9} + 973144 p^{2} T^{10} - 119702 p^{3} T^{11} + 17239 p^{4} T^{12} - 1586 p^{5} T^{13} + 187 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 14 T + 293 T^{2} + 2740 T^{3} + 33615 T^{4} + 242402 T^{5} + 2242224 T^{6} + 13275744 T^{7} + 99943923 T^{8} + 13275744 p T^{9} + 2242224 p^{2} T^{10} + 242402 p^{3} T^{11} + 33615 p^{4} T^{12} + 2740 p^{5} T^{13} + 293 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 22 T + 364 T^{2} + 3988 T^{3} + 38788 T^{4} + 316942 T^{5} + 2547011 T^{6} + 18287930 T^{7} + 125620167 T^{8} + 18287930 p T^{9} + 2547011 p^{2} T^{10} + 316942 p^{3} T^{11} + 38788 p^{4} T^{12} + 3988 p^{5} T^{13} + 364 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 14 T + 4 p T^{2} + 1202 T^{3} + 206 p T^{4} + 59172 T^{5} + 521607 T^{6} + 4081424 T^{7} + 30196845 T^{8} + 4081424 p T^{9} + 521607 p^{2} T^{10} + 59172 p^{3} T^{11} + 206 p^{5} T^{12} + 1202 p^{5} T^{13} + 4 p^{7} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 280 T^{2} - 2648 T^{3} + 36932 T^{4} - 323816 T^{5} + 3034841 T^{6} - 23656728 T^{7} + 170490407 T^{8} - 23656728 p T^{9} + 3034841 p^{2} T^{10} - 323816 p^{3} T^{11} + 36932 p^{4} T^{12} - 2648 p^{5} T^{13} + 280 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 18 T + 350 T^{2} + 3744 T^{3} + 44214 T^{4} + 359832 T^{5} + 3454539 T^{6} + 24383320 T^{7} + 206612515 T^{8} + 24383320 p T^{9} + 3454539 p^{2} T^{10} + 359832 p^{3} T^{11} + 44214 p^{4} T^{12} + 3744 p^{5} T^{13} + 350 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 T + 299 T^{2} - 474 T^{3} + 45285 T^{4} - 60926 T^{5} + 4462108 T^{6} - 5123706 T^{7} + 309748025 T^{8} - 5123706 p T^{9} + 4462108 p^{2} T^{10} - 60926 p^{3} T^{11} + 45285 p^{4} T^{12} - 474 p^{5} T^{13} + 299 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 14 T + 329 T^{2} - 54 p T^{3} + 42816 T^{4} - 337200 T^{5} + 3175907 T^{6} - 22237940 T^{7} + 190023299 T^{8} - 22237940 p T^{9} + 3175907 p^{2} T^{10} - 337200 p^{3} T^{11} + 42816 p^{4} T^{12} - 54 p^{6} T^{13} + 329 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 10 T + 363 T^{2} - 3452 T^{3} + 65848 T^{4} - 571942 T^{5} + 7668881 T^{6} - 57892138 T^{7} + 613900435 T^{8} - 57892138 p T^{9} + 7668881 p^{2} T^{10} - 571942 p^{3} T^{11} + 65848 p^{4} T^{12} - 3452 p^{5} T^{13} + 363 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 2 T + 169 T^{2} - 952 T^{3} + 15013 T^{4} - 2478 p T^{5} + 1444406 T^{6} - 17048980 T^{7} + 114845337 T^{8} - 17048980 p T^{9} + 1444406 p^{2} T^{10} - 2478 p^{4} T^{11} + 15013 p^{4} T^{12} - 952 p^{5} T^{13} + 169 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 16 T + 427 T^{2} + 3862 T^{3} + 59688 T^{4} + 272670 T^{5} + 3713457 T^{6} + 487666 T^{7} + 178388803 T^{8} + 487666 p T^{9} + 3713457 p^{2} T^{10} + 272670 p^{3} T^{11} + 59688 p^{4} T^{12} + 3862 p^{5} T^{13} + 427 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 290 T^{2} - 1724 T^{3} + 13131 T^{4} + 106136 T^{5} - 7564 p T^{6} + 15893784 T^{7} - 44516827 T^{8} + 15893784 p T^{9} - 7564 p^{3} T^{10} + 106136 p^{3} T^{11} + 13131 p^{4} T^{12} - 1724 p^{5} T^{13} + 290 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 46 T + 1199 T^{2} + 23144 T^{3} + 369333 T^{4} + 5064130 T^{5} + 60930310 T^{6} + 651823514 T^{7} + 6260522181 T^{8} + 651823514 p T^{9} + 60930310 p^{2} T^{10} + 5064130 p^{3} T^{11} + 369333 p^{4} T^{12} + 23144 p^{5} T^{13} + 1199 p^{6} T^{14} + 46 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 38 T + 1150 T^{2} - 22786 T^{3} + 390897 T^{4} - 5296342 T^{5} + 65582590 T^{6} - 692743768 T^{7} + 6971358239 T^{8} - 692743768 p T^{9} + 65582590 p^{2} T^{10} - 5296342 p^{3} T^{11} + 390897 p^{4} T^{12} - 22786 p^{5} T^{13} + 1150 p^{6} T^{14} - 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 4 T + 372 T^{2} + 2788 T^{3} + 79078 T^{4} + 643798 T^{5} + 12404343 T^{6} + 89443170 T^{7} + 1412723929 T^{8} + 89443170 p T^{9} + 12404343 p^{2} T^{10} + 643798 p^{3} T^{11} + 79078 p^{4} T^{12} + 2788 p^{5} T^{13} + 372 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.55132956922488185625661811382, −3.49393407967379481907829488582, −3.46750455627154999987913302958, −3.36346609761522622571406743551, −3.26096373148554855695672178063, −3.24443655032619481375607007964, −3.11796742844885354663612765858, −3.07402001758763996458933035666, −2.80859692671299058767601295235, −2.65481410718057026953825061598, −2.49117928694064696206761568527, −2.37137752656288858130963650090, −2.35709281585516250231458644982, −2.30723758268921041078096679043, −2.20172417058435687340411300438, −1.96408019759743233090842600511, −1.96373093320871666731923724602, −1.94506499917752255694545523059, −1.35935714794482561497546812193, −1.34291201171161684247745558790, −1.33361711372540164113322051757, −1.31119115966279728506136636598, −1.17231364599469402066425997924, −1.05661651759851793204932662248, −1.00354310364536232533104100791, 0, 0, 0, 0, 0, 0, 0, 0, 1.00354310364536232533104100791, 1.05661651759851793204932662248, 1.17231364599469402066425997924, 1.31119115966279728506136636598, 1.33361711372540164113322051757, 1.34291201171161684247745558790, 1.35935714794482561497546812193, 1.94506499917752255694545523059, 1.96373093320871666731923724602, 1.96408019759743233090842600511, 2.20172417058435687340411300438, 2.30723758268921041078096679043, 2.35709281585516250231458644982, 2.37137752656288858130963650090, 2.49117928694064696206761568527, 2.65481410718057026953825061598, 2.80859692671299058767601295235, 3.07402001758763996458933035666, 3.11796742844885354663612765858, 3.24443655032619481375607007964, 3.26096373148554855695672178063, 3.36346609761522622571406743551, 3.46750455627154999987913302958, 3.49393407967379481907829488582, 3.55132956922488185625661811382
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.