Dirichlet series
| L(s) = 1 | + 2·2-s + 4·4-s + 7-s + 2·8-s − 11-s − 2·13-s + 2·14-s + 16-s − 22·17-s + 4·19-s − 2·22-s + 15·23-s − 4·26-s + 4·28-s + 29-s + 4·31-s − 6·32-s − 44·34-s − 2·37-s + 8·38-s − 5·41-s + 10·43-s − 4·44-s + 30·46-s + 20·47-s + 16·49-s − 8·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2·4-s + 0.377·7-s + 0.707·8-s − 0.301·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 5.33·17-s + 0.917·19-s − 0.426·22-s + 3.12·23-s − 0.784·26-s + 0.755·28-s + 0.185·29-s + 0.718·31-s − 1.06·32-s − 7.54·34-s − 0.328·37-s + 1.29·38-s − 0.780·41-s + 1.52·43-s − 0.603·44-s + 4.42·46-s + 2.91·47-s + 16/7·49-s − 1.10·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{24} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(712273.\) |
| Root analytic conductor: | \(2.32161\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{24} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.186959684\) |
| \(L(\frac12)\) | \(\approx\) | \(7.186959684\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 2 | \( 1 - p T + 3 p T^{3} - 9 T^{4} + p T^{5} + 9 T^{6} - 13 T^{7} + 7 T^{8} - 13 p T^{9} + 9 p^{2} T^{10} + p^{4} T^{11} - 9 p^{4} T^{12} + 3 p^{6} T^{13} - p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - T - 15 T^{2} - 22 T^{3} + 142 T^{4} + 303 T^{5} - 274 T^{6} - 1448 T^{7} - 633 T^{8} - 1448 p T^{9} - 274 p^{2} T^{10} + 303 p^{3} T^{11} + 142 p^{4} T^{12} - 22 p^{5} T^{13} - 15 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + T - 18 T^{2} - 129 T^{3} + 120 T^{4} + 1886 T^{5} + 6087 T^{6} - 14944 T^{7} - 88007 T^{8} - 14944 p T^{9} + 6087 p^{2} T^{10} + 1886 p^{3} T^{11} + 120 p^{4} T^{12} - 129 p^{5} T^{13} - 18 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T - 18 T^{2} - 102 T^{3} - 3 T^{4} + 1191 T^{5} + 3365 T^{6} - 5033 T^{7} - 45777 T^{8} - 5033 p T^{9} + 3365 p^{2} T^{10} + 1191 p^{3} T^{11} - 3 p^{4} T^{12} - 102 p^{5} T^{13} - 18 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 + 11 T + 88 T^{2} + 451 T^{3} + 2111 T^{4} + 451 p T^{5} + 88 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 34 p T^{5} + 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 15 T + 76 T^{2} - 183 T^{3} + 1258 T^{4} - 9882 T^{5} + 28951 T^{6} - 29994 T^{7} + 75241 T^{8} - 29994 p T^{9} + 28951 p^{2} T^{10} - 9882 p^{3} T^{11} + 1258 p^{4} T^{12} - 183 p^{5} T^{13} + 76 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - T - 75 T^{2} - 186 T^{3} + 3234 T^{4} + 10969 T^{5} - 64920 T^{6} - 187910 T^{7} + 1140565 T^{8} - 187910 p T^{9} - 64920 p^{2} T^{10} + 10969 p^{3} T^{11} + 3234 p^{4} T^{12} - 186 p^{5} T^{13} - 75 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 4 T - 66 T^{2} + 362 T^{3} + 1939 T^{4} - 11745 T^{5} - 45361 T^{6} + 144883 T^{7} + 1472277 T^{8} + 144883 p T^{9} - 45361 p^{2} T^{10} - 11745 p^{3} T^{11} + 1939 p^{4} T^{12} + 362 p^{5} T^{13} - 66 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 + T + 49 T^{2} - 392 T^{3} + 241 T^{4} - 392 p T^{5} + 49 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 5 T - 114 T^{2} - 213 T^{3} + 9222 T^{4} + 3430 T^{5} - 12387 p T^{6} - 107426 T^{7} + 20984173 T^{8} - 107426 p T^{9} - 12387 p^{3} T^{10} + 3430 p^{3} T^{11} + 9222 p^{4} T^{12} - 213 p^{5} T^{13} - 114 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 10 T - 24 T^{2} + 168 T^{3} + 2043 T^{4} + 8292 T^{5} - 128884 T^{6} - 10646 T^{7} + 2342736 T^{8} - 10646 p T^{9} - 128884 p^{2} T^{10} + 8292 p^{3} T^{11} + 2043 p^{4} T^{12} + 168 p^{5} T^{13} - 24 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 20 T + 105 T^{2} - 120 T^{3} + 6294 T^{4} - 71140 T^{5} + 174165 T^{6} - 1240450 T^{7} + 19222555 T^{8} - 1240450 p T^{9} + 174165 p^{2} T^{10} - 71140 p^{3} T^{11} + 6294 p^{4} T^{12} - 120 p^{5} T^{13} + 105 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 + 20 T + 298 T^{2} + 3001 T^{3} + 25499 T^{4} + 3001 p T^{5} + 298 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 - 17 T + 51 T^{2} + 144 T^{3} + 2220 T^{4} + 17291 T^{5} - 531846 T^{6} + 1855556 T^{7} + 4239943 T^{8} + 1855556 p T^{9} - 531846 p^{2} T^{10} + 17291 p^{3} T^{11} + 2220 p^{4} T^{12} + 144 p^{5} T^{13} + 51 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 13 T - 72 T^{2} + 1443 T^{3} + 6378 T^{4} - 120042 T^{5} - 78037 T^{6} + 2420938 T^{7} + 9620433 T^{8} + 2420938 p T^{9} - 78037 p^{2} T^{10} - 120042 p^{3} T^{11} + 6378 p^{4} T^{12} + 1443 p^{5} T^{13} - 72 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 17 T - 15 T^{2} - 1066 T^{3} + 7426 T^{4} + 68373 T^{5} - 833566 T^{6} + 1085128 T^{7} + 111947307 T^{8} + 1085128 p T^{9} - 833566 p^{2} T^{10} + 68373 p^{3} T^{11} + 7426 p^{4} T^{12} - 1066 p^{5} T^{13} - 15 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 8 T + 244 T^{2} - 1441 T^{3} + 24947 T^{4} - 1441 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 2 T + 196 T^{2} - 679 T^{3} + 18071 T^{4} - 679 p T^{5} + 196 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( 1 - 7 T - 234 T^{2} + 1199 T^{3} + 36520 T^{4} - 121434 T^{5} - 3999001 T^{6} + 3702556 T^{7} + 362101113 T^{8} + 3702556 p T^{9} - 3999001 p^{2} T^{10} - 121434 p^{3} T^{11} + 36520 p^{4} T^{12} + 1199 p^{5} T^{13} - 234 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 30 T + 280 T^{2} - 1770 T^{3} + 42079 T^{4} - 523485 T^{5} + 2681215 T^{6} - 35481675 T^{7} + 533008645 T^{8} - 35481675 p T^{9} + 2681215 p^{2} T^{10} - 523485 p^{3} T^{11} + 42079 p^{4} T^{12} - 1770 p^{5} T^{13} + 280 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 9 T + 257 T^{2} - 1998 T^{3} + 31929 T^{4} - 1998 p T^{5} + 257 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 19 T - 108 T^{2} + 1825 T^{3} + 482 p T^{4} - 332118 T^{5} - 5651993 T^{6} + 4946342 T^{7} + 777287673 T^{8} + 4946342 p T^{9} - 5651993 p^{2} T^{10} - 332118 p^{3} T^{11} + 482 p^{5} T^{12} + 1825 p^{5} T^{13} - 108 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.67424529596192635529559634724, −4.26726916748150760318806178598, −4.22143161909657374090227309186, −4.19529225265420572270845069496, −4.05159814834040768740864101118, −4.04090680532616469938729663250, −3.78262639940932781363633442390, −3.66689312471558008549950241864, −3.31523592606609423583759737796, −3.26352676261105135842224032789, −3.10613465705889698943306200679, −3.05858940551987778118822409961, −2.83340900830698967671131051181, −2.50892798407847778650053865894, −2.50046094299090050520872735498, −2.44563123761985560045033282626, −2.38136835969692106340247236280, −2.19108602951918060473080479406, −1.78216406749349795545343190122, −1.74609353651287406568692646933, −1.73238277175492281038185802128, −1.00604477808104059259474071811, −0.832939376532421252641380507555, −0.813393206786288813169961555785, −0.27196629254392591602233281685, 0.27196629254392591602233281685, 0.813393206786288813169961555785, 0.832939376532421252641380507555, 1.00604477808104059259474071811, 1.73238277175492281038185802128, 1.74609353651287406568692646933, 1.78216406749349795545343190122, 2.19108602951918060473080479406, 2.38136835969692106340247236280, 2.44563123761985560045033282626, 2.50046094299090050520872735498, 2.50892798407847778650053865894, 2.83340900830698967671131051181, 3.05858940551987778118822409961, 3.10613465705889698943306200679, 3.26352676261105135842224032789, 3.31523592606609423583759737796, 3.66689312471558008549950241864, 3.78262639940932781363633442390, 4.04090680532616469938729663250, 4.05159814834040768740864101118, 4.19529225265420572270845069496, 4.22143161909657374090227309186, 4.26726916748150760318806178598, 4.67424529596192635529559634724
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.