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
−4.35700334228156191177975374346, −4.34797542282241678296371789210, −4.22236541335488726787408480973, −4.06406061164483134410964251451, −3.91168129084516564887464935529, −3.79120314296459164973731016058, −3.68891547011130357371936929907, −3.59100762900733806563108325083, −3.44947334884715984770508653217, −3.32415437551241882489602903441, −3.28754706918599082200172057705, −2.78990631505765155350396978147, −2.78823388648393718032501100768, −2.59822715710740763693091224356, −2.38021449095739538253880278496, −2.37083798407704941980822581505, −2.22800276076251992325285286081, −1.93951300528619820219998305722, −1.50793206761887389993996670212, −1.45397331447750814175227502396, −1.37835583599496737815986596781, −1.30214759933018876267573995103, −0.76114127874463738391218017558, −0.64744647800304890071788867040, −0.63364421515237526450281278724, 0.63364421515237526450281278724, 0.64744647800304890071788867040, 0.76114127874463738391218017558, 1.30214759933018876267573995103, 1.37835583599496737815986596781, 1.45397331447750814175227502396, 1.50793206761887389993996670212, 1.93951300528619820219998305722, 2.22800276076251992325285286081, 2.37083798407704941980822581505, 2.38021449095739538253880278496, 2.59822715710740763693091224356, 2.78823388648393718032501100768, 2.78990631505765155350396978147, 3.28754706918599082200172057705, 3.32415437551241882489602903441, 3.44947334884715984770508653217, 3.59100762900733806563108325083, 3.68891547011130357371936929907, 3.79120314296459164973731016058, 3.91168129084516564887464935529, 4.06406061164483134410964251451, 4.22236541335488726787408480973, 4.34797542282241678296371789210, 4.35700334228156191177975374346
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.