Dirichlet series
| L(s) = 1 | − 8·3-s + 2·5-s + 7-s + 36·9-s + 8·11-s − 16·15-s + 4·17-s − 5·19-s − 8·21-s + 23-s − 17·25-s − 120·27-s − 2·29-s + 9·31-s − 64·33-s + 2·35-s − 5·37-s + 11·41-s + 6·43-s + 72·45-s + 11·47-s + 18·49-s − 32·51-s + 40·53-s + 16·55-s + 40·57-s + 28·59-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 0.894·5-s + 0.377·7-s + 12·9-s + 2.41·11-s − 4.13·15-s + 0.970·17-s − 1.14·19-s − 1.74·21-s + 0.208·23-s − 3.39·25-s − 23.0·27-s − 0.371·29-s + 1.61·31-s − 11.1·33-s + 0.338·35-s − 0.821·37-s + 1.71·41-s + 0.914·43-s + 10.7·45-s + 1.60·47-s + 18/7·49-s − 4.48·51-s + 5.49·53-s + 2.15·55-s + 5.29·57-s + 3.64·59-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
| \(L(1)\) | \(\approx\) | \(1.623308867\) |
| \(L(\frac12)\) | \(\approx\) | \(1.623308867\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 33 T + 545 T^{2} + 6105 T^{3} + 54459 T^{4} + 6105 p T^{5} + 545 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( ( 1 - T + 2 p T^{2} - 12 T^{3} + 62 T^{4} - 12 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - T - 17 T^{2} + 18 T^{3} + 20 p T^{4} - 118 T^{5} - 1026 T^{6} + 323 T^{7} + 7697 T^{8} + 323 p T^{9} - 1026 p^{2} T^{10} - 118 p^{3} T^{11} + 20 p^{5} T^{12} + 18 p^{5} T^{13} - 17 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 13 | \( 1 - 23 T^{2} - 6 T^{3} + 94 T^{4} + 108 T^{5} - 2222 T^{6} - 489 T^{7} + 71146 T^{8} - 489 p T^{9} - 2222 p^{2} T^{10} + 108 p^{3} T^{11} + 94 p^{4} T^{12} - 6 p^{5} T^{13} - 23 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 4 T - 29 T^{2} + 126 T^{3} + 311 T^{4} - 670 T^{5} - 10575 T^{6} - 364 p T^{7} + 304921 T^{8} - 364 p^{2} T^{9} - 10575 p^{2} T^{10} - 670 p^{3} T^{11} + 311 p^{4} T^{12} + 126 p^{5} T^{13} - 29 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T - 25 T^{2} - 242 T^{3} - 48 T^{4} + 3794 T^{5} + 7346 T^{6} - 18627 T^{7} - 102551 T^{8} - 18627 p T^{9} + 7346 p^{2} T^{10} + 3794 p^{3} T^{11} - 48 p^{4} T^{12} - 242 p^{5} T^{13} - 25 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - T - 24 T^{2} + 31 T^{3} + 279 T^{4} - 708 T^{5} + 18640 T^{6} - 2096 T^{7} - 503040 T^{8} - 2096 p T^{9} + 18640 p^{2} T^{10} - 708 p^{3} T^{11} + 279 p^{4} T^{12} + 31 p^{5} T^{13} - 24 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T - 43 T^{2} - 96 T^{3} - 115 T^{4} - 1316 T^{5} - 17729 T^{6} + 71592 T^{7} + 1922293 T^{8} + 71592 p T^{9} - 17729 p^{2} T^{10} - 1316 p^{3} T^{11} - 115 p^{4} T^{12} - 96 p^{5} T^{13} - 43 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 9 T + 21 T^{2} + 334 T^{3} - 3603 T^{4} + 19093 T^{5} - 11834 T^{6} - 590409 T^{7} + 5215882 T^{8} - 590409 p T^{9} - 11834 p^{2} T^{10} + 19093 p^{3} T^{11} - 3603 p^{4} T^{12} + 334 p^{5} T^{13} + 21 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 5 T - 23 T^{2} + 836 T^{3} + 4710 T^{4} - 16170 T^{5} + 319326 T^{6} + 1913289 T^{7} - 4942379 T^{8} + 1913289 p T^{9} + 319326 p^{2} T^{10} - 16170 p^{3} T^{11} + 4710 p^{4} T^{12} + 836 p^{5} T^{13} - 23 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 11 T - 5 T^{2} - 66 T^{3} + 3342 T^{4} + 5808 T^{5} - 102972 T^{6} + 250283 T^{7} - 2878959 T^{8} + 250283 p T^{9} - 102972 p^{2} T^{10} + 5808 p^{3} T^{11} + 3342 p^{4} T^{12} - 66 p^{5} T^{13} - 5 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 3 T + 57 T^{2} + 133 T^{3} + 1551 T^{4} + 133 p T^{5} + 57 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 11 T - 29 T^{2} + 66 T^{3} + 108 p T^{4} + 6666 T^{5} - 242622 T^{6} + 458909 T^{7} - 280527 T^{8} + 458909 p T^{9} - 242622 p^{2} T^{10} + 6666 p^{3} T^{11} + 108 p^{5} T^{12} + 66 p^{5} T^{13} - 29 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 - 20 T + 333 T^{2} - 3393 T^{3} + 29770 T^{4} - 3393 p T^{5} + 333 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( ( 1 - 14 T + 201 T^{2} - 1563 T^{3} + 14548 T^{4} - 1563 p T^{5} + 201 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 - 20 T + 241 T^{2} - 2946 T^{3} + 27920 T^{4} - 224900 T^{5} + 1845888 T^{6} - 13520291 T^{7} + 97623770 T^{8} - 13520291 p T^{9} + 1845888 p^{2} T^{10} - 224900 p^{3} T^{11} + 27920 p^{4} T^{12} - 2946 p^{5} T^{13} + 241 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 11 T + 54 T^{2} + 1111 T^{3} - 19219 T^{4} + 170456 T^{5} - 99822 T^{6} - 11375958 T^{7} + 161395572 T^{8} - 11375958 p T^{9} - 99822 p^{2} T^{10} + 170456 p^{3} T^{11} - 19219 p^{4} T^{12} + 1111 p^{5} T^{13} + 54 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 7 T - 130 T^{2} - 2149 T^{3} + 4271 T^{4} + 206570 T^{5} + 1092495 T^{6} - 8070615 T^{7} - 120604568 T^{8} - 8070615 p T^{9} + 1092495 p^{2} T^{10} + 206570 p^{3} T^{11} + 4271 p^{4} T^{12} - 2149 p^{5} T^{13} - 130 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 12 T - 47 T^{2} - 486 T^{3} + 12854 T^{4} + 63174 T^{5} - 99654 T^{6} - 4520379 T^{7} - 25440400 T^{8} - 4520379 p T^{9} - 99654 p^{2} T^{10} + 63174 p^{3} T^{11} + 12854 p^{4} T^{12} - 486 p^{5} T^{13} - 47 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 4 T - 293 T^{2} - 642 T^{3} + 54065 T^{4} + 66808 T^{5} - 6809565 T^{6} - 2000374 T^{7} + 658163011 T^{8} - 2000374 p T^{9} - 6809565 p^{2} T^{10} + 66808 p^{3} T^{11} + 54065 p^{4} T^{12} - 642 p^{5} T^{13} - 293 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 8 T + 295 T^{2} + 1841 T^{3} + 36468 T^{4} + 1841 p T^{5} + 295 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 20 T - 53 T^{2} + 1854 T^{3} + 28472 T^{4} - 297470 T^{5} - 2904618 T^{6} + 7196245 T^{7} + 391603694 T^{8} + 7196245 p T^{9} - 2904618 p^{2} T^{10} - 297470 p^{3} T^{11} + 28472 p^{4} T^{12} + 1854 p^{5} T^{13} - 53 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.50787884572904726950799130310, −4.23323900443912059626398912708, −4.14012478527300088394886741299, −4.02460548060863016419149939512, −3.99292109078751081083609052827, −3.88209008676129061588437495665, −3.78460784232369220577977908816, −3.77762922228934395036255792250, −3.77481962775732220629674275137, −3.29562754982060815426501113775, −3.13605519773129455404671977901, −2.66838496804971466250752631435, −2.39741867783611540227180623641, −2.38898790673550169901712074357, −2.33024770271164299396128969893, −2.28556635728315413208164275526, −2.20601558774055147003288484817, −1.61724318612866070566807204850, −1.45689771046797655523194934587, −1.34788290326371809416923571549, −1.17785955714102836814123734224, −1.07543074553081861579739367971, −0.799225912235688098073033755799, −0.50112103640509532582389824494, −0.37889026753825062223188651647, 0.37889026753825062223188651647, 0.50112103640509532582389824494, 0.799225912235688098073033755799, 1.07543074553081861579739367971, 1.17785955714102836814123734224, 1.34788290326371809416923571549, 1.45689771046797655523194934587, 1.61724318612866070566807204850, 2.20601558774055147003288484817, 2.28556635728315413208164275526, 2.33024770271164299396128969893, 2.38898790673550169901712074357, 2.39741867783611540227180623641, 2.66838496804971466250752631435, 3.13605519773129455404671977901, 3.29562754982060815426501113775, 3.77481962775732220629674275137, 3.77762922228934395036255792250, 3.78460784232369220577977908816, 3.88209008676129061588437495665, 3.99292109078751081083609052827, 4.02460548060863016419149939512, 4.14012478527300088394886741299, 4.23323900443912059626398912708, 4.50787884572904726950799130310
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.