Dirichlet series
| L(s) = 1 | + 8·3-s − 6·5-s + 36·9-s + 3·11-s − 2·13-s − 48·15-s − 17-s + 4·19-s + 12·23-s + 7·25-s + 120·27-s − 9·29-s − 31-s + 24·33-s + 14·37-s − 16·39-s + 10·41-s + 8·43-s − 216·45-s + 16·47-s + 17·49-s − 8·51-s − 12·53-s − 18·55-s + 32·57-s + 4·61-s + 12·65-s + ⋯ |
| L(s) = 1 | + 4.61·3-s − 2.68·5-s + 12·9-s + 0.904·11-s − 0.554·13-s − 12.3·15-s − 0.242·17-s + 0.917·19-s + 2.50·23-s + 7/5·25-s + 23.0·27-s − 1.67·29-s − 0.179·31-s + 4.17·33-s + 2.30·37-s − 2.56·39-s + 1.56·41-s + 1.21·43-s − 32.1·45-s + 2.33·47-s + 17/7·49-s − 1.12·51-s − 1.64·53-s − 2.42·55-s + 4.23·57-s + 0.512·61-s + 1.48·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 67^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 67^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{8} \cdot 67^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.88579\times 10^{6}\) |
| Root analytic conductor: | \(2.53376\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{8} \cdot 67^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(43.48848546\) |
| \(L(\frac12)\) | \(\approx\) | \(43.48848546\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 67 | \( 1 - 20 T + 301 T^{2} - 3485 T^{3} + 29812 T^{4} - 3485 p T^{5} + 301 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( ( 1 + 3 T + 2 p T^{2} + 22 T^{3} + 56 T^{4} + 22 p T^{5} + 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 17 T^{2} + 2 p T^{3} + 150 T^{4} - 24 p T^{5} - 648 T^{6} + 99 p T^{7} + 2416 T^{8} + 99 p^{2} T^{9} - 648 p^{2} T^{10} - 24 p^{4} T^{11} + 150 p^{4} T^{12} + 2 p^{6} T^{13} - 17 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 - 3 T - 13 T^{2} + 4 p T^{3} + 5 T^{4} + 69 T^{5} - 34 p T^{6} - 239 p T^{7} + 16074 T^{8} - 239 p^{2} T^{9} - 34 p^{3} T^{10} + 69 p^{3} T^{11} + 5 p^{4} T^{12} + 4 p^{6} T^{13} - 13 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T - 17 T^{2} + 56 T^{3} + 242 T^{4} - 1090 T^{5} + 2856 T^{6} + 13823 T^{7} - 40782 T^{8} + 13823 p T^{9} + 2856 p^{2} T^{10} - 1090 p^{3} T^{11} + 242 p^{4} T^{12} + 56 p^{5} T^{13} - 17 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + T - 60 T^{2} - 25 T^{3} + 2179 T^{4} + 356 T^{5} - 3239 p T^{6} - 2183 T^{7} + 1066504 T^{8} - 2183 p T^{9} - 3239 p^{3} T^{10} + 356 p^{3} T^{11} + 2179 p^{4} T^{12} - 25 p^{5} T^{13} - 60 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T - 33 T^{2} + 246 T^{3} + 294 T^{4} - 276 p T^{5} + 7496 T^{6} + 43027 T^{7} - 209844 T^{8} + 43027 p T^{9} + 7496 p^{2} T^{10} - 276 p^{4} T^{11} + 294 p^{4} T^{12} + 246 p^{5} T^{13} - 33 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 29 | \( 1 + 9 T - 52 T^{2} - 349 T^{3} + 4715 T^{4} + 16920 T^{5} - 180779 T^{6} - 85603 T^{7} + 7243236 T^{8} - 85603 p T^{9} - 180779 p^{2} T^{10} + 16920 p^{3} T^{11} + 4715 p^{4} T^{12} - 349 p^{5} T^{13} - 52 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + T - 59 T^{2} - 414 T^{3} + 1629 T^{4} + 18099 T^{5} + 60102 T^{6} - 383231 T^{7} - 3058022 T^{8} - 383231 p T^{9} + 60102 p^{2} T^{10} + 18099 p^{3} T^{11} + 1629 p^{4} T^{12} - 414 p^{5} T^{13} - 59 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 14 T + 69 T^{2} - 320 T^{3} + 1338 T^{4} + 8824 T^{5} - 80750 T^{6} + 365453 T^{7} - 2739878 T^{8} + 365453 p T^{9} - 80750 p^{2} T^{10} + 8824 p^{3} T^{11} + 1338 p^{4} T^{12} - 320 p^{5} T^{13} + 69 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 10 T - 49 T^{2} + 596 T^{3} + 2552 T^{4} - 13790 T^{5} - 212474 T^{6} - 65943 T^{7} + 14026710 T^{8} - 65943 p T^{9} - 212474 p^{2} T^{10} - 13790 p^{3} T^{11} + 2552 p^{4} T^{12} + 596 p^{5} T^{13} - 49 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 165 T^{2} - 489 T^{3} + 10476 T^{4} - 489 p T^{5} + 165 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 16 T - 15 T^{2} + 490 T^{3} + 13708 T^{4} - 91334 T^{5} - 519604 T^{6} - 745255 T^{7} + 58651408 T^{8} - 745255 p T^{9} - 519604 p^{2} T^{10} - 91334 p^{3} T^{11} + 13708 p^{4} T^{12} + 490 p^{5} T^{13} - 15 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 + 6 T + 73 T^{2} + 691 T^{3} + 5558 T^{4} + 691 p T^{5} + 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( ( 1 + 203 T^{2} - 55 T^{3} + 17052 T^{4} - 55 p T^{5} + 203 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 - 4 T - 107 T^{2} - 178 T^{3} + 5888 T^{4} + 656 p T^{5} - 157692 T^{6} - 1734955 T^{7} + 9802722 T^{8} - 1734955 p T^{9} - 157692 p^{2} T^{10} + 656 p^{4} T^{11} + 5888 p^{4} T^{12} - 178 p^{5} T^{13} - 107 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 6 T - 160 T^{2} + 920 T^{3} + 13043 T^{4} - 50796 T^{5} - 1112732 T^{6} + 1091174 T^{7} + 92960496 T^{8} + 1091174 p T^{9} - 1112732 p^{2} T^{10} - 50796 p^{3} T^{11} + 13043 p^{4} T^{12} + 920 p^{5} T^{13} - 160 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 13 T - 38 T^{2} - 1143 T^{3} - 4293 T^{4} - 40986 T^{5} - 537885 T^{6} + 4059643 T^{7} + 100253128 T^{8} + 4059643 p T^{9} - 537885 p^{2} T^{10} - 40986 p^{3} T^{11} - 4293 p^{4} T^{12} - 1143 p^{5} T^{13} - 38 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 18 T - 61 T^{2} - 1412 T^{3} + 24774 T^{4} + 193916 T^{5} - 2351956 T^{6} + 166685 T^{7} + 321032056 T^{8} + 166685 p T^{9} - 2351956 p^{2} T^{10} + 193916 p^{3} T^{11} + 24774 p^{4} T^{12} - 1412 p^{5} T^{13} - 61 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 15 T - 142 T^{2} - 1553 T^{3} + 34937 T^{4} + 183540 T^{5} - 4065077 T^{6} - 3275795 T^{7} + 440135994 T^{8} - 3275795 p T^{9} - 4065077 p^{2} T^{10} + 183540 p^{3} T^{11} + 34937 p^{4} T^{12} - 1553 p^{5} T^{13} - 142 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 2 T + 193 T^{2} - 325 T^{3} + 19010 T^{4} - 325 p T^{5} + 193 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 30 T + 257 T^{2} - 980 T^{3} + 33000 T^{4} - 426760 T^{5} + 1244192 T^{6} - 35456635 T^{7} + 727608454 T^{8} - 35456635 p T^{9} + 1244192 p^{2} T^{10} - 426760 p^{3} T^{11} + 33000 p^{4} T^{12} - 980 p^{5} T^{13} + 257 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.35669749076072457604687954826, −4.15811436701670817650613240426, −3.95417035603408264771270861149, −3.93455677306927594591628525886, −3.77073022293339161083754292009, −3.74329127591001364234658271584, −3.68060005154061841702382557668, −3.49409570345034926835385128310, −3.47118456104803723380917941711, −3.27073528904870619947061063516, −2.91664129839871667917216460638, −2.84620767045306958396545920163, −2.72887036527465866463131544906, −2.62515591209729302214247745263, −2.55031559987718918785842062702, −2.40636374514142685198843112305, −2.39876506677986799349090875808, −1.85790977956359539436599445564, −1.83077572389203743802212411173, −1.70626605116071304402635488218, −1.34231007164417011734035656919, −1.30530155213276258816773888159, −0.72595133083739669909869142151, −0.71384860107970799001348880857, −0.62652900817684408342945057155, 0.62652900817684408342945057155, 0.71384860107970799001348880857, 0.72595133083739669909869142151, 1.30530155213276258816773888159, 1.34231007164417011734035656919, 1.70626605116071304402635488218, 1.83077572389203743802212411173, 1.85790977956359539436599445564, 2.39876506677986799349090875808, 2.40636374514142685198843112305, 2.55031559987718918785842062702, 2.62515591209729302214247745263, 2.72887036527465866463131544906, 2.84620767045306958396545920163, 2.91664129839871667917216460638, 3.27073528904870619947061063516, 3.47118456104803723380917941711, 3.49409570345034926835385128310, 3.68060005154061841702382557668, 3.74329127591001364234658271584, 3.77073022293339161083754292009, 3.93455677306927594591628525886, 3.95417035603408264771270861149, 4.15811436701670817650613240426, 4.35669749076072457604687954826
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.