Dirichlet series
| L(s) = 1 | − 2·5-s − 3·7-s + 5·11-s − 8·13-s − 6·17-s + 6·19-s − 30·23-s + 25-s − 9·31-s + 6·35-s − 7·37-s − 25·41-s − 8·43-s + 7·47-s + 6·49-s − 3·53-s − 10·55-s − 5·59-s + 13·61-s + 16·65-s + 18·67-s − 5·71-s − 15·73-s − 15·77-s − 50·79-s − 13·83-s + 12·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 1.50·11-s − 2.21·13-s − 1.45·17-s + 1.37·19-s − 6.25·23-s + 1/5·25-s − 1.61·31-s + 1.01·35-s − 1.15·37-s − 3.90·41-s − 1.21·43-s + 1.02·47-s + 6/7·49-s − 0.412·53-s − 1.34·55-s − 0.650·59-s + 1.66·61-s + 1.98·65-s + 2.19·67-s − 0.593·71-s − 1.75·73-s − 1.70·77-s − 5.62·79-s − 1.42·83-s + 1.30·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.90425\times 10^{9}\) |
| Root analytic conductor: | \(3.97622\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01983962131\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01983962131\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 5 T + 26 T^{2} - 75 T^{3} + 251 T^{4} - 75 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( 1 + 3 T + 3 T^{2} + 12 T^{3} + 11 p T^{4} + 72 T^{5} - 6 T^{6} + 669 T^{7} + 3697 T^{8} + 669 p T^{9} - 6 p^{2} T^{10} + 72 p^{3} T^{11} + 11 p^{5} T^{12} + 12 p^{5} T^{13} + 3 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 8 T - 9 T^{2} - 222 T^{3} - 146 T^{4} + 314 p T^{5} + 12777 T^{6} - 19546 T^{7} - 202497 T^{8} - 19546 p T^{9} + 12777 p^{2} T^{10} + 314 p^{4} T^{11} - 146 p^{4} T^{12} - 222 p^{5} T^{13} - 9 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T + 7 T^{2} - 10 T^{3} + 230 T^{4} + 564 T^{5} - 4629 T^{6} - 28410 T^{7} - 78541 T^{8} - 28410 p T^{9} - 4629 p^{2} T^{10} + 564 p^{3} T^{11} + 230 p^{4} T^{12} - 10 p^{5} T^{13} + 7 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 6 T + 59 T^{2} - 446 T^{3} + 2866 T^{4} - 832 p T^{5} + 91851 T^{6} - 429198 T^{7} + 1967627 T^{8} - 429198 p T^{9} + 91851 p^{2} T^{10} - 832 p^{4} T^{11} + 2866 p^{4} T^{12} - 446 p^{5} T^{13} + 59 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 15 T + 147 T^{2} + 1010 T^{3} + 5579 T^{4} + 1010 p T^{5} + 147 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 43 T^{2} - 360 T^{3} + 558 T^{4} + 18000 T^{5} + 63169 T^{6} - 254250 T^{7} - 3533245 T^{8} - 254250 p T^{9} + 63169 p^{2} T^{10} + 18000 p^{3} T^{11} + 558 p^{4} T^{12} - 360 p^{5} T^{13} - 43 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 + 9 T + p T^{2} + 33 T^{3} - 953 T^{4} - 286 p T^{5} + 3562 T^{6} + 290518 T^{7} + 1969841 T^{8} + 290518 p T^{9} + 3562 p^{2} T^{10} - 286 p^{4} T^{11} - 953 p^{4} T^{12} + 33 p^{5} T^{13} + p^{7} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 7 T - 45 T^{2} - 159 T^{3} + 3601 T^{4} + 10648 T^{5} - 110484 T^{6} + 57880 T^{7} + 6304899 T^{8} + 57880 p T^{9} - 110484 p^{2} T^{10} + 10648 p^{3} T^{11} + 3601 p^{4} T^{12} - 159 p^{5} T^{13} - 45 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 25 T + 233 T^{2} + 890 T^{3} + 383 T^{4} - 12100 T^{5} - 177984 T^{6} - 1888625 T^{7} - 13773645 T^{8} - 1888625 p T^{9} - 177984 p^{2} T^{10} - 12100 p^{3} T^{11} + 383 p^{4} T^{12} + 890 p^{5} T^{13} + 233 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 4 T + 100 T^{2} + 249 T^{3} + 5381 T^{4} + 249 p T^{5} + 100 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 7 T - 115 T^{2} + 1024 T^{3} + 2531 T^{4} - 65408 T^{5} + 352806 T^{6} + 1510805 T^{7} - 29832051 T^{8} + 1510805 p T^{9} + 352806 p^{2} T^{10} - 65408 p^{3} T^{11} + 2531 p^{4} T^{12} + 1024 p^{5} T^{13} - 115 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 3 T - 58 T^{2} - 406 T^{3} - 994 T^{4} + 59053 T^{5} + 336110 T^{6} - 1679748 T^{7} - 16721881 T^{8} - 1679748 p T^{9} + 336110 p^{2} T^{10} + 59053 p^{3} T^{11} - 994 p^{4} T^{12} - 406 p^{5} T^{13} - 58 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 5 T - 109 T^{2} - 645 T^{3} + 3865 T^{4} + 41460 T^{5} + 200474 T^{6} - 1266670 T^{7} - 29160191 T^{8} - 1266670 p T^{9} + 200474 p^{2} T^{10} + 41460 p^{3} T^{11} + 3865 p^{4} T^{12} - 645 p^{5} T^{13} - 109 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 13 T - 28 T^{2} + 1097 T^{3} - 2572 T^{4} - 53888 T^{5} + 332010 T^{6} + 1938280 T^{7} - 32800569 T^{8} + 1938280 p T^{9} + 332010 p^{2} T^{10} - 53888 p^{3} T^{11} - 2572 p^{4} T^{12} + 1097 p^{5} T^{13} - 28 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 9 T + 254 T^{2} - 1733 T^{3} + 25029 T^{4} - 1733 p T^{5} + 254 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 5 T - 157 T^{2} - 935 T^{3} + 7583 T^{4} + 74830 T^{5} + 532206 T^{6} - 2313250 T^{7} - 81202695 T^{8} - 2313250 p T^{9} + 532206 p^{2} T^{10} + 74830 p^{3} T^{11} + 7583 p^{4} T^{12} - 935 p^{5} T^{13} - 157 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 15 T - 42 T^{2} - 1315 T^{3} + 1410 T^{4} + 91730 T^{5} + 456748 T^{6} - 2965680 T^{7} - 62115581 T^{8} - 2965680 p T^{9} + 456748 p^{2} T^{10} + 91730 p^{3} T^{11} + 1410 p^{4} T^{12} - 1315 p^{5} T^{13} - 42 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( ( 1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 1505 p T^{5} + 231 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 83 | \( 1 + 13 T - 139 T^{2} - 2887 T^{3} - 3721 T^{4} + 283622 T^{5} + 2834202 T^{6} - 10613696 T^{7} - 346217067 T^{8} - 10613696 p T^{9} + 2834202 p^{2} T^{10} + 283622 p^{3} T^{11} - 3721 p^{4} T^{12} - 2887 p^{5} T^{13} - 139 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 16 T + 327 T^{2} + 3278 T^{3} + 41595 T^{4} + 3278 p T^{5} + 327 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - T + 92 T^{2} - 280 T^{3} + 17090 T^{4} - 50219 T^{5} + 1379316 T^{6} - 4543970 T^{7} + 231425799 T^{8} - 4543970 p T^{9} + 1379316 p^{2} T^{10} - 50219 p^{3} T^{11} + 17090 p^{4} T^{12} - 280 p^{5} T^{13} + 92 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.74816838932580240407793319487, −3.74001213797885012220629025380, −3.70781793788488401179658173866, −3.66534883743508866544807357578, −3.58401320302501860974336502146, −3.11427846165194022853019284652, −3.09310718836792628657973942959, −3.04957946765303483923848567860, −2.86612916062127367389833051668, −2.61680254487576798961840317215, −2.61538522672079041266212167956, −2.49433392087472498202338784546, −2.41851696351318545496447916678, −2.08050131254427150981638929134, −1.89944169602615210040058858993, −1.88910863632093026689920656782, −1.80223234627957857734446166845, −1.49124869194379007220174294383, −1.47429505770888693522996806774, −1.37779846418504372954455028123, −1.28646645614749908221299166502, −0.59820503320968194454837677032, −0.31301741807189864234727191987, −0.27037478095586645051124469124, −0.04681583418460220292706391386, 0.04681583418460220292706391386, 0.27037478095586645051124469124, 0.31301741807189864234727191987, 0.59820503320968194454837677032, 1.28646645614749908221299166502, 1.37779846418504372954455028123, 1.47429505770888693522996806774, 1.49124869194379007220174294383, 1.80223234627957857734446166845, 1.88910863632093026689920656782, 1.89944169602615210040058858993, 2.08050131254427150981638929134, 2.41851696351318545496447916678, 2.49433392087472498202338784546, 2.61538522672079041266212167956, 2.61680254487576798961840317215, 2.86612916062127367389833051668, 3.04957946765303483923848567860, 3.09310718836792628657973942959, 3.11427846165194022853019284652, 3.58401320302501860974336502146, 3.66534883743508866544807357578, 3.70781793788488401179658173866, 3.74001213797885012220629025380, 3.74816838932580240407793319487
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.