Dirichlet series
| L(s) = 1 | + 8·3-s − 4·4-s + 8·5-s − 8·7-s − 16·8-s + 28·9-s − 32·12-s + 64·15-s + 8·16-s + 32·19-s − 32·20-s − 64·21-s − 32·23-s − 128·24-s + 24·25-s − 8·27-s + 32·28-s − 24·31-s + 56·32-s − 64·35-s − 112·36-s + 56·37-s − 128·40-s + 56·41-s − 96·43-s + 224·45-s − 80·47-s + ⋯ |
| L(s) = 1 | + 8/3·3-s − 4-s + 8/5·5-s − 8/7·7-s − 2·8-s + 28/9·9-s − 8/3·12-s + 4.26·15-s + 1/2·16-s + 1.68·19-s − 8/5·20-s − 3.04·21-s − 1.39·23-s − 5.33·24-s + 0.959·25-s − 0.296·27-s + 8/7·28-s − 0.774·31-s + 7/4·32-s − 1.82·35-s − 3.11·36-s + 1.51·37-s − 3.19·40-s + 1.36·41-s − 2.23·43-s + 4.97·45-s − 1.70·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.47864\times 10^{7}\) |
| Root analytic conductor: | \(2.80618\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 17^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.7926797590\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7926797590\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T^{2} + p^{4} T^{3} + p^{3} T^{4} + 9 p^{3} T^{5} + 37 p^{2} T^{6} + 25 p^{3} T^{7} + 801 T^{8} + 25 p^{5} T^{9} + 37 p^{6} T^{10} + 9 p^{9} T^{11} + p^{11} T^{12} + p^{14} T^{13} + p^{14} T^{14} + p^{16} T^{16} \) |
| 3 | \( 1 - 8 T + 4 p^{2} T^{2} - 56 T^{3} - 164 T^{4} + 352 p T^{5} - 1180 T^{6} - 10096 T^{7} + 48536 T^{8} - 10096 p^{2} T^{9} - 1180 p^{4} T^{10} + 352 p^{7} T^{11} - 164 p^{8} T^{12} - 56 p^{10} T^{13} + 4 p^{14} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 5 | \( 1 - 8 T + 8 p T^{2} - 344 T^{3} + 98 p^{2} T^{4} - 488 p^{2} T^{5} + 68376 T^{6} - 377784 T^{7} + 1765218 T^{8} - 377784 p^{2} T^{9} + 68376 p^{4} T^{10} - 488 p^{8} T^{11} + 98 p^{10} T^{12} - 344 p^{10} T^{13} + 8 p^{13} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 + 8 T + 36 T^{2} + 296 T^{3} - 1188 T^{4} + 8480 T^{5} + 50404 T^{6} - 323328 T^{7} + 5214232 T^{8} - 323328 p^{2} T^{9} + 50404 p^{4} T^{10} + 8480 p^{6} T^{11} - 1188 p^{8} T^{12} + 296 p^{10} T^{13} + 36 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 100 T^{2} + 168 p T^{3} + 23412 T^{4} - 32640 T^{5} + 5079196 T^{6} + 24153816 T^{7} + 171031352 T^{8} + 24153816 p^{2} T^{9} + 5079196 p^{4} T^{10} - 32640 p^{6} T^{11} + 23412 p^{8} T^{12} + 168 p^{11} T^{13} + 100 p^{12} T^{14} + p^{16} T^{16} \) | |
| 13 | \( 1 + 784 T^{3} + 9888 T^{4} - 237552 T^{5} + 307328 T^{6} - 20917120 T^{7} + 236770498 T^{8} - 20917120 p^{2} T^{9} + 307328 p^{4} T^{10} - 237552 p^{6} T^{11} + 9888 p^{8} T^{12} + 784 p^{10} T^{13} + p^{16} T^{16} \) | |
| 19 | \( 1 - 32 T + 544 T^{2} - 10608 T^{3} + 192512 T^{4} - 3096432 T^{5} + 96494752 T^{6} - 2593739296 T^{7} + 50479240962 T^{8} - 2593739296 p^{2} T^{9} + 96494752 p^{4} T^{10} - 3096432 p^{6} T^{11} + 192512 p^{8} T^{12} - 10608 p^{10} T^{13} + 544 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 32 T - 188 T^{2} - 48872 T^{3} - 828684 T^{4} + 13728208 T^{5} + 720883772 T^{6} - 931890616 T^{7} - 277592396616 T^{8} - 931890616 p^{2} T^{9} + 720883772 p^{4} T^{10} + 13728208 p^{6} T^{11} - 828684 p^{8} T^{12} - 48872 p^{10} T^{13} - 188 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 396 T^{2} - 6480 T^{3} - 41850 T^{4} - 12650832 T^{5} + 327376404 T^{6} + 8912834592 T^{7} - 17449190910 T^{8} + 8912834592 p^{2} T^{9} + 327376404 p^{4} T^{10} - 12650832 p^{6} T^{11} - 41850 p^{8} T^{12} - 6480 p^{10} T^{13} - 396 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( 1 + 24 T - 1188 T^{2} + 53744 T^{3} + 2074612 T^{4} - 73705208 T^{5} + 1616756820 T^{6} + 68616424944 T^{7} - 2275613089160 T^{8} + 68616424944 p^{2} T^{9} + 1616756820 p^{4} T^{10} - 73705208 p^{6} T^{11} + 2074612 p^{8} T^{12} + 53744 p^{10} T^{13} - 1188 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 56 T + 2440 T^{2} - 35624 T^{3} - 2601518 T^{4} + 151524328 T^{5} - 4208954696 T^{6} + 29245838776 T^{7} + 3837195651554 T^{8} + 29245838776 p^{2} T^{9} - 4208954696 p^{4} T^{10} + 151524328 p^{6} T^{11} - 2601518 p^{8} T^{12} - 35624 p^{10} T^{13} + 2440 p^{12} T^{14} - 56 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 56 T - 3828 T^{2} + 333784 T^{3} + 43462 T^{4} - 795405800 T^{5} + 27829322796 T^{6} + 654539020488 T^{7} - 72848043583742 T^{8} + 654539020488 p^{2} T^{9} + 27829322796 p^{4} T^{10} - 795405800 p^{6} T^{11} + 43462 p^{8} T^{12} + 333784 p^{10} T^{13} - 3828 p^{12} T^{14} - 56 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 96 T + 6096 T^{2} + 290832 T^{3} + 11553408 T^{4} + 238904592 T^{5} - 4015115952 T^{6} - 725328019488 T^{7} - 38391385271230 T^{8} - 725328019488 p^{2} T^{9} - 4015115952 p^{4} T^{10} + 238904592 p^{6} T^{11} + 11553408 p^{8} T^{12} + 290832 p^{10} T^{13} + 6096 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 80 T + 3200 T^{2} + 228944 T^{3} + 25698900 T^{4} + 1284226160 T^{5} + 46709290368 T^{6} + 3079489449072 T^{7} + 201735188722534 T^{8} + 3079489449072 p^{2} T^{9} + 46709290368 p^{4} T^{10} + 1284226160 p^{6} T^{11} + 25698900 p^{8} T^{12} + 228944 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 96 T + 3844 T^{2} + 250080 T^{3} + 17203208 T^{4} + 878356128 T^{5} + 61303275468 T^{6} + 2343880946208 T^{7} + 44401903321166 T^{8} + 2343880946208 p^{2} T^{9} + 61303275468 p^{4} T^{10} + 878356128 p^{6} T^{11} + 17203208 p^{8} T^{12} + 250080 p^{10} T^{13} + 3844 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 8 T - 1064 T^{2} - 139576 T^{3} - 8160 p T^{4} - 260011896 T^{5} + 38349218008 T^{6} + 28053020936 T^{7} + 74913788078050 T^{8} + 28053020936 p^{2} T^{9} + 38349218008 p^{4} T^{10} - 260011896 p^{6} T^{11} - 8160 p^{9} T^{12} - 139576 p^{10} T^{13} - 1064 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 56 T + 3112 T^{2} - 6296 T^{3} - 15027214 T^{4} + 501099544 T^{5} - 20121844008 T^{6} - 701499756936 T^{7} + 88775644719138 T^{8} - 701499756936 p^{2} T^{9} - 20121844008 p^{4} T^{10} + 501099544 p^{6} T^{11} - 15027214 p^{8} T^{12} - 6296 p^{10} T^{13} + 3112 p^{12} T^{14} - 56 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 30264 T^{2} + 421009100 T^{4} - 3513162425416 T^{6} + 19237575139662822 T^{8} - 3513162425416 p^{4} T^{10} + 421009100 p^{8} T^{12} - 30264 p^{12} T^{14} + p^{16} T^{16} \) | |
| 71 | \( 1 - 136 T + 11956 T^{2} - 1645208 T^{3} + 125751260 T^{4} - 6420023472 T^{5} + 608611686116 T^{6} - 36734819612880 T^{7} + 1344565035705944 T^{8} - 36734819612880 p^{2} T^{9} + 608611686116 p^{4} T^{10} - 6420023472 p^{6} T^{11} + 125751260 p^{8} T^{12} - 1645208 p^{10} T^{13} + 11956 p^{12} T^{14} - 136 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 - 240 T + 25588 T^{2} - 800272 T^{3} - 116781882 T^{4} + 16237112912 T^{5} - 532918753260 T^{6} - 64355395952464 T^{7} + 8703183849681538 T^{8} - 64355395952464 p^{2} T^{9} - 532918753260 p^{4} T^{10} + 16237112912 p^{6} T^{11} - 116781882 p^{8} T^{12} - 800272 p^{10} T^{13} + 25588 p^{12} T^{14} - 240 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 328 T + 69692 T^{2} - 11584672 T^{3} + 1597267636 T^{4} - 188486263336 T^{5} + 19641637188212 T^{6} - 1821893660223504 T^{7} + 151643860636286456 T^{8} - 1821893660223504 p^{2} T^{9} + 19641637188212 p^{4} T^{10} - 188486263336 p^{6} T^{11} + 1597267636 p^{8} T^{12} - 11584672 p^{10} T^{13} + 69692 p^{12} T^{14} - 328 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 88 T + 12840 T^{2} + 772808 T^{3} - 70014880 T^{4} + 15998431560 T^{5} + 276363214376 T^{6} - 33752671318936 T^{7} + 12884842311213474 T^{8} - 33752671318936 p^{2} T^{9} + 276363214376 p^{4} T^{10} + 15998431560 p^{6} T^{11} - 70014880 p^{8} T^{12} + 772808 p^{10} T^{13} + 12840 p^{12} T^{14} - 88 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 288 T + 41472 T^{2} - 4656976 T^{3} + 519733472 T^{4} - 56560463824 T^{5} + 5578739762816 T^{6} - 499331992822432 T^{7} + 43753371662175234 T^{8} - 499331992822432 p^{2} T^{9} + 5578739762816 p^{4} T^{10} - 56560463824 p^{6} T^{11} + 519733472 p^{8} T^{12} - 4656976 p^{10} T^{13} + 41472 p^{12} T^{14} - 288 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 280 T + 28216 T^{2} + 294440 T^{3} - 455917166 T^{4} + 59371138584 T^{5} - 2146688572344 T^{6} - 4011403168744 p T^{7} + 7060740010466 p^{2} T^{8} - 4011403168744 p^{3} T^{9} - 2146688572344 p^{4} T^{10} + 59371138584 p^{6} T^{11} - 455917166 p^{8} T^{12} + 294440 p^{10} T^{13} + 28216 p^{12} T^{14} - 280 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−5.11909240570704019449379627010, −5.05664659259999664109827752923, −4.77410934562591646234430591970, −4.66777036660051871635237695708, −4.36615100866436826660106909255, −3.95697845728656556263853276533, −3.92350484319478637631604611251, −3.69102915530328342037303896134, −3.68059814986523193302071796159, −3.59668682601131905982249350504, −3.49795069782804220340554621769, −3.38139317229959439709722396902, −3.05856314403634684971374532065, −2.85625255378875600006245321338, −2.76286011636066380193842933595, −2.54036457315779879156662344902, −2.45560647910272705719155466444, −2.11400390412292775907101733238, −1.97345495366454934501227228275, −1.89011242915344643135375630364, −1.73639954638680637438609166045, −1.18113352031691835199557483404, −0.839539418565869640349383529032, −0.61087284470856454942872053128, −0.084408206868584840951903245693, 0.084408206868584840951903245693, 0.61087284470856454942872053128, 0.839539418565869640349383529032, 1.18113352031691835199557483404, 1.73639954638680637438609166045, 1.89011242915344643135375630364, 1.97345495366454934501227228275, 2.11400390412292775907101733238, 2.45560647910272705719155466444, 2.54036457315779879156662344902, 2.76286011636066380193842933595, 2.85625255378875600006245321338, 3.05856314403634684971374532065, 3.38139317229959439709722396902, 3.49795069782804220340554621769, 3.59668682601131905982249350504, 3.68059814986523193302071796159, 3.69102915530328342037303896134, 3.92350484319478637631604611251, 3.95697845728656556263853276533, 4.36615100866436826660106909255, 4.66777036660051871635237695708, 4.77410934562591646234430591970, 5.05664659259999664109827752923, 5.11909240570704019449379627010
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.