Dirichlet series
| L(s) = 1 | + 2·5-s − 4·7-s + 11-s − 20·13-s + 4·17-s + 19-s − 12·23-s + 5·25-s − 12·29-s + 8·31-s − 8·35-s + 12·41-s + 2·43-s + 9·47-s + 11·49-s − 7·53-s + 2·55-s + 4·59-s − 25·61-s − 40·65-s − 30·67-s + 22·71-s + 4·73-s − 4·77-s + 7·79-s − 58·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.301·11-s − 5.54·13-s + 0.970·17-s + 0.229·19-s − 2.50·23-s + 25-s − 2.22·29-s + 1.43·31-s − 1.35·35-s + 1.87·41-s + 0.304·43-s + 1.31·47-s + 11/7·49-s − 0.961·53-s + 0.269·55-s + 0.520·59-s − 3.20·61-s − 4.96·65-s − 3.66·67-s + 2.61·71-s + 0.468·73-s − 0.455·77-s + 0.787·79-s − 6.36·83-s + 0.867·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.51472\times 10^{8}\) |
| Root analytic conductor: | \(3.47467\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1357927861\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1357927861\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 7 | \( 1 + 4 T + 5 T^{2} - 33 T^{3} - 136 T^{4} - 33 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 2 T - T^{2} - 4 p T^{3} + 36 T^{4} + 46 T^{5} + 66 p T^{6} - 151 p T^{7} - 374 T^{8} - 151 p^{2} T^{9} + 66 p^{3} T^{10} + 46 p^{3} T^{11} + 36 p^{4} T^{12} - 4 p^{6} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - T - 15 T^{2} - 108 T^{3} + 240 T^{4} + 1366 T^{5} + 5502 T^{6} - 16607 T^{7} - 62387 T^{8} - 16607 p T^{9} + 5502 p^{2} T^{10} + 1366 p^{3} T^{11} + 240 p^{4} T^{12} - 108 p^{5} T^{13} - 15 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 + 10 T + 73 T^{2} + 389 T^{3} + 1556 T^{4} + 389 p T^{5} + 73 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 - 4 T - 29 T^{2} + 246 T^{3} + 203 T^{4} - 5446 T^{5} + 13245 T^{6} + 47608 T^{7} - 341903 T^{8} + 47608 p T^{9} + 13245 p^{2} T^{10} - 5446 p^{3} T^{11} + 203 p^{4} T^{12} + 246 p^{5} T^{13} - 29 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - T + 3 T^{2} + 94 T^{3} - 40 T^{4} - 786 T^{5} - 206 T^{6} - 9541 T^{7} - 158247 T^{8} - 9541 p T^{9} - 206 p^{2} T^{10} - 786 p^{3} T^{11} - 40 p^{4} T^{12} + 94 p^{5} T^{13} + 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 12 T + 61 T^{2} + 194 T^{3} + 271 T^{4} - 838 T^{5} - 11959 T^{6} - 133924 T^{7} - 856813 T^{8} - 133924 p T^{9} - 11959 p^{2} T^{10} - 838 p^{3} T^{11} + 271 p^{4} T^{12} + 194 p^{5} T^{13} + 61 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 + 6 T + 59 T^{2} + 387 T^{3} + 2091 T^{4} + 387 p T^{5} + 59 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 - 8 T + 9 T^{2} + 102 T^{3} - 1287 T^{4} + 10998 T^{5} - 5977 T^{6} - 318082 T^{7} + 2302767 T^{8} - 318082 p T^{9} - 5977 p^{2} T^{10} + 10998 p^{3} T^{11} - 1287 p^{4} T^{12} + 102 p^{5} T^{13} + 9 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 63 T^{2} - 114 T^{3} + 1781 T^{4} + 5700 T^{5} + 37899 T^{6} - 141018 T^{7} - 2640591 T^{8} - 141018 p T^{9} + 37899 p^{2} T^{10} + 5700 p^{3} T^{11} + 1781 p^{4} T^{12} - 114 p^{5} T^{13} - 63 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 6 T + 127 T^{2} - 453 T^{3} + 6650 T^{4} - 453 p T^{5} + 127 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - T + 144 T^{2} - 50 T^{3} + 8638 T^{4} - 50 p T^{5} + 144 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 9 T - 67 T^{2} + 660 T^{3} + 2564 T^{4} - 7452 T^{5} - 311076 T^{6} - 354939 T^{7} + 24590333 T^{8} - 354939 p T^{9} - 311076 p^{2} T^{10} - 7452 p^{3} T^{11} + 2564 p^{4} T^{12} + 660 p^{5} T^{13} - 67 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 7 T - 158 T^{2} - 729 T^{3} + 18893 T^{4} + 54478 T^{5} - 1436103 T^{6} - 995821 T^{7} + 90249700 T^{8} - 995821 p T^{9} - 1436103 p^{2} T^{10} + 54478 p^{3} T^{11} + 18893 p^{4} T^{12} - 729 p^{5} T^{13} - 158 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 4 T - 21 T^{2} + 738 T^{3} - 6108 T^{4} + 3112 T^{5} + 138210 T^{6} - 1929881 T^{7} + 10657612 T^{8} - 1929881 p T^{9} + 138210 p^{2} T^{10} + 3112 p^{3} T^{11} - 6108 p^{4} T^{12} + 738 p^{5} T^{13} - 21 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 25 T + 196 T^{2} + 1097 T^{3} + 19941 T^{4} + 171360 T^{5} + 257114 T^{6} + 4239618 T^{7} + 85475992 T^{8} + 4239618 p T^{9} + 257114 p^{2} T^{10} + 171360 p^{3} T^{11} + 19941 p^{4} T^{12} + 1097 p^{5} T^{13} + 196 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 30 T + 5 p T^{2} + 2816 T^{3} + 39697 T^{4} + 415264 T^{5} + 2662719 T^{6} + 27218576 T^{7} + 300754815 T^{8} + 27218576 p T^{9} + 2662719 p^{2} T^{10} + 415264 p^{3} T^{11} + 39697 p^{4} T^{12} + 2816 p^{5} T^{13} + 5 p^{7} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 11 T + 194 T^{2} - 1324 T^{3} + 16844 T^{4} - 1324 p T^{5} + 194 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 4 T - 177 T^{2} + 378 T^{3} + 16890 T^{4} - 8478 T^{5} - 1265668 T^{6} + 148369 T^{7} + 87542238 T^{8} + 148369 p T^{9} - 1265668 p^{2} T^{10} - 8478 p^{3} T^{11} + 16890 p^{4} T^{12} + 378 p^{5} T^{13} - 177 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 7 T + 6 T^{2} + 609 T^{3} - 7809 T^{4} + 35112 T^{5} + 213797 T^{6} - 5183759 T^{7} + 23372886 T^{8} - 5183759 p T^{9} + 213797 p^{2} T^{10} + 35112 p^{3} T^{11} - 7809 p^{4} T^{12} + 609 p^{5} T^{13} + 6 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 29 T + 560 T^{2} + 7232 T^{3} + 75754 T^{4} + 7232 p T^{5} + 560 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 9 T - 128 T^{2} + 549 T^{3} + 20701 T^{4} - 107280 T^{5} - 329006 T^{6} + 7780266 T^{7} - 45486752 T^{8} + 7780266 p T^{9} - 329006 p^{2} T^{10} - 107280 p^{3} T^{11} + 20701 p^{4} T^{12} + 549 p^{5} T^{13} - 128 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 + 4 T + 192 T^{2} - 412 T^{3} + 15886 T^{4} - 412 p T^{5} + 192 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−3.94264273614852086687627037531, −3.90696200893927589429050960213, −3.86128369527737805081271873833, −3.84660062345302518684917080427, −3.45005141234438452511195762171, −3.34697309509325148818584609938, −3.05022498803608971695829725126, −2.89910646818735846884281196447, −2.85873634737547917703270796811, −2.84764526198768120201437054545, −2.69524914439081703605454820465, −2.68840482779247008945762116530, −2.47507232106565847224502883662, −2.37232114338093050156854234112, −2.16755956619563328627590355335, −2.07561076705768704329819370517, −1.92774144951291400983682395395, −1.68315398748542842305828104739, −1.40892832868903719392089337375, −1.22228501766281975872403110307, −1.19107242590372255888706191673, −1.17925045362973926136832937671, −0.35009753323871427038345690404, −0.22216158605547059221214188696, −0.12559548770747369883499609350, 0.12559548770747369883499609350, 0.22216158605547059221214188696, 0.35009753323871427038345690404, 1.17925045362973926136832937671, 1.19107242590372255888706191673, 1.22228501766281975872403110307, 1.40892832868903719392089337375, 1.68315398748542842305828104739, 1.92774144951291400983682395395, 2.07561076705768704329819370517, 2.16755956619563328627590355335, 2.37232114338093050156854234112, 2.47507232106565847224502883662, 2.68840482779247008945762116530, 2.69524914439081703605454820465, 2.84764526198768120201437054545, 2.85873634737547917703270796811, 2.89910646818735846884281196447, 3.05022498803608971695829725126, 3.34697309509325148818584609938, 3.45005141234438452511195762171, 3.84660062345302518684917080427, 3.86128369527737805081271873833, 3.90696200893927589429050960213, 3.94264273614852086687627037531
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.