Dirichlet series
| L(s) = 1 | + 8·2-s + 32·4-s + 80·8-s + 16·11-s − 32·13-s + 120·16-s − 16·17-s + 128·22-s + 64·23-s − 256·26-s − 48·31-s + 32·32-s − 128·34-s − 80·37-s + 128·41-s + 32·43-s + 512·44-s + 512·46-s + 160·47-s − 1.02e3·52-s − 136·53-s + 16·61-s − 384·62-s − 384·64-s − 64·67-s − 512·68-s − 240·71-s + ⋯ |
| L(s) = 1 | + 4·2-s + 8·4-s + 10·8-s + 1.45·11-s − 2.46·13-s + 15/2·16-s − 0.941·17-s + 5.81·22-s + 2.78·23-s − 9.84·26-s − 1.54·31-s + 32-s − 3.76·34-s − 2.16·37-s + 3.12·41-s + 0.744·43-s + 11.6·44-s + 11.1·46-s + 3.40·47-s − 19.6·52-s − 2.56·53-s + 0.262·61-s − 6.19·62-s − 6·64-s − 0.955·67-s − 7.52·68-s − 3.38·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48946\times 10^{11}\) |
| Root analytic conductor: | \(5.34887\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(39.15115728\) |
| \(L(\frac12)\) | \(\approx\) | \(39.15115728\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| good | 11 | \( ( 1 - 8 T + 312 T^{2} - 2152 T^{3} + 50354 T^{4} - 2152 p^{2} T^{5} + 312 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 32 T + 512 T^{2} + 6560 T^{3} + 134332 T^{4} + 2762336 T^{5} + 41133568 T^{6} + 495718368 T^{7} + 5952416454 T^{8} + 495718368 p^{2} T^{9} + 41133568 p^{4} T^{10} + 2762336 p^{6} T^{11} + 134332 p^{8} T^{12} + 6560 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 16 T + 128 T^{2} + 1808 T^{3} + 15620 T^{4} + 449840 T^{5} + 6832512 T^{6} + 61777968 T^{7} - 1397251322 T^{8} + 61777968 p^{2} T^{9} + 6832512 p^{4} T^{10} + 449840 p^{6} T^{11} + 15620 p^{8} T^{12} + 1808 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 1584 T^{2} + 1414948 T^{4} - 831048336 T^{6} + 352477351302 T^{8} - 831048336 p^{4} T^{10} + 1414948 p^{8} T^{12} - 1584 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 - 64 T + 2048 T^{2} - 72256 T^{3} + 3035716 T^{4} - 89425600 T^{5} + 2116556800 T^{6} - 58954795200 T^{7} + 1580427481350 T^{8} - 58954795200 p^{2} T^{9} + 2116556800 p^{4} T^{10} - 89425600 p^{6} T^{11} + 3035716 p^{8} T^{12} - 72256 p^{10} T^{13} + 2048 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 3064 T^{2} + 4188508 T^{4} - 3369785416 T^{6} + 2445991211782 T^{8} - 3369785416 p^{4} T^{10} + 4188508 p^{8} T^{12} - 3064 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 + 24 T + 1080 T^{2} + 41016 T^{3} + 1877138 T^{4} + 41016 p^{2} T^{5} + 1080 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 + 80 T + 3200 T^{2} + 194384 T^{3} + 5899396 T^{4} - 73090192 T^{5} - 5832712832 T^{6} - 463726994064 T^{7} - 29036398330170 T^{8} - 463726994064 p^{2} T^{9} - 5832712832 p^{4} T^{10} - 73090192 p^{6} T^{11} + 5899396 p^{8} T^{12} + 194384 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 - 64 T + 7452 T^{2} - 302528 T^{3} + 19058438 T^{4} - 302528 p^{2} T^{5} + 7452 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 - 32 T + 512 T^{2} + 77920 T^{3} - 842108 T^{4} - 91679456 T^{5} + 6400665088 T^{6} + 184942936992 T^{7} - 17230346966586 T^{8} + 184942936992 p^{2} T^{9} + 6400665088 p^{4} T^{10} - 91679456 p^{6} T^{11} - 842108 p^{8} T^{12} + 77920 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 160 T + 12800 T^{2} - 882976 T^{3} + 57817156 T^{4} - 3159456352 T^{5} + 155276727808 T^{6} - 7818642741984 T^{7} + 384595118070150 T^{8} - 7818642741984 p^{2} T^{9} + 155276727808 p^{4} T^{10} - 3159456352 p^{6} T^{11} + 57817156 p^{8} T^{12} - 882976 p^{10} T^{13} + 12800 p^{12} T^{14} - 160 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 136 T + 9248 T^{2} + 610648 T^{3} + 21977980 T^{4} - 392622200 T^{5} - 70203488288 T^{6} - 6434028915432 T^{7} - 455550832999482 T^{8} - 6434028915432 p^{2} T^{9} - 70203488288 p^{4} T^{10} - 392622200 p^{6} T^{11} + 21977980 p^{8} T^{12} + 610648 p^{10} T^{13} + 9248 p^{12} T^{14} + 136 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 21672 T^{2} + 221983516 T^{4} - 1395223762968 T^{6} + 5869610712107526 T^{8} - 1395223762968 p^{4} T^{10} + 221983516 p^{8} T^{12} - 21672 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 8 T + 6876 T^{2} + 71816 T^{3} + 26974310 T^{4} + 71816 p^{2} T^{5} + 6876 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 + 64 T + 2048 T^{2} + 278080 T^{3} + 11734852 T^{4} - 454571840 T^{5} - 3221504 p^{2} T^{6} - 31219859904 p T^{7} - 301847111409210 T^{8} - 31219859904 p^{3} T^{9} - 3221504 p^{6} T^{10} - 454571840 p^{6} T^{11} + 11734852 p^{8} T^{12} + 278080 p^{10} T^{13} + 2048 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 120 T + 19224 T^{2} + 1354200 T^{3} + 132459410 T^{4} + 1354200 p^{2} T^{5} + 19224 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 181440 T^{3} - 8077700 T^{4} - 1465853760 T^{5} + 16460236800 T^{6} - 4077836784000 T^{7} + 224471955023238 T^{8} - 4077836784000 p^{2} T^{9} + 16460236800 p^{4} T^{10} - 1465853760 p^{6} T^{11} - 8077700 p^{8} T^{12} + 181440 p^{10} T^{13} + p^{16} T^{16} \) | |
| 79 | \( 1 - 25448 T^{2} + 278884572 T^{4} - 1882085710552 T^{6} + 11110496353393094 T^{8} - 1882085710552 p^{4} T^{10} + 278884572 p^{8} T^{12} - 25448 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 + 32 T + 512 T^{2} + 160160 T^{3} - 117869948 T^{4} - 2250812704 T^{5} + 1149019648 T^{6} + 2913519278688 T^{7} + 7271273442846534 T^{8} + 2913519278688 p^{2} T^{9} + 1149019648 p^{4} T^{10} - 2250812704 p^{6} T^{11} - 117869948 p^{8} T^{12} + 160160 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 27464 T^{2} + 430658844 T^{4} - 4752521547256 T^{6} + 41804566556072390 T^{8} - 4752521547256 p^{4} T^{10} + 430658844 p^{8} T^{12} - 27464 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 - 128 T + 8192 T^{2} - 819520 T^{3} + 178540796 T^{4} - 19975944256 T^{5} + 1430121179136 T^{6} - 145069633254912 T^{7} + 13571651589735430 T^{8} - 145069633254912 p^{2} T^{9} + 1430121179136 p^{4} T^{10} - 19975944256 p^{6} T^{11} + 178540796 p^{8} T^{12} - 819520 p^{10} T^{13} + 8192 p^{12} T^{14} - 128 p^{14} T^{15} + p^{16} T^{16} \) | |
| 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
−4.10384731727175796209092982468, −3.95495109815957901669763473623, −3.87840213889432315482099304430, −3.67051626284326448679338592618, −3.60697888394006327173939807914, −3.36860909132964683344275850234, −3.18200995155019473708225818018, −3.13730968758196231672754671554, −3.11862759489679842178298155128, −2.99138678133425375995908315862, −2.70076685678843812529596550331, −2.65587207689788643868245857033, −2.44781121335025670904407906211, −2.28815394252498332673433753705, −2.22953329722494534261558760550, −2.20474977843406267832036698612, −1.88249635619744223203481021737, −1.54770099072672212163177660545, −1.54081173431644385612373803772, −1.37700459476153512648586803894, −1.16782602896196858579323373308, −0.71075590149404032163372399607, −0.62820587279779330271557575687, −0.46468411431842841299413656381, −0.15320854688798521422485504930, 0.15320854688798521422485504930, 0.46468411431842841299413656381, 0.62820587279779330271557575687, 0.71075590149404032163372399607, 1.16782602896196858579323373308, 1.37700459476153512648586803894, 1.54081173431644385612373803772, 1.54770099072672212163177660545, 1.88249635619744223203481021737, 2.20474977843406267832036698612, 2.22953329722494534261558760550, 2.28815394252498332673433753705, 2.44781121335025670904407906211, 2.65587207689788643868245857033, 2.70076685678843812529596550331, 2.99138678133425375995908315862, 3.11862759489679842178298155128, 3.13730968758196231672754671554, 3.18200995155019473708225818018, 3.36860909132964683344275850234, 3.60697888394006327173939807914, 3.67051626284326448679338592618, 3.87840213889432315482099304430, 3.95495109815957901669763473623, 4.10384731727175796209092982468
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.