Dirichlet series
| L(s) = 1 | + 2·2-s + 8·3-s − 3·4-s + 2·5-s + 16·6-s + 8·7-s − 6·8-s + 36·9-s + 4·10-s + 8·11-s − 24·12-s + 16·14-s + 16·15-s + 8·16-s + 10·17-s + 72·18-s + 18·19-s − 6·20-s + 64·21-s + 16·22-s − 2·23-s − 48·24-s − 21·25-s + 120·27-s − 24·28-s − 12·29-s + 32·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4.61·3-s − 3/2·4-s + 0.894·5-s + 6.53·6-s + 3.02·7-s − 2.12·8-s + 12·9-s + 1.26·10-s + 2.41·11-s − 6.92·12-s + 4.27·14-s + 4.13·15-s + 2·16-s + 2.42·17-s + 16.9·18-s + 4.12·19-s − 1.34·20-s + 13.9·21-s + 3.41·22-s − 0.417·23-s − 9.79·24-s − 4.19·25-s + 23.0·27-s − 4.53·28-s − 2.22·29-s + 5.84·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 7^{8} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.15972\times 10^{11}\) |
| Root analytic conductor: | \(5.32343\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 7^{8} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2092.024047\) |
| \(L(\frac12)\) | \(\approx\) | \(2092.024047\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{8} \) |
| 7 | \( ( 1 - T )^{8} \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + 7 T^{2} - 7 p T^{3} + 29 T^{4} - 13 p^{2} T^{5} + 23 p^{2} T^{6} - 17 p^{3} T^{7} + 217 T^{8} - 17 p^{4} T^{9} + 23 p^{4} T^{10} - 13 p^{5} T^{11} + 29 p^{4} T^{12} - 7 p^{6} T^{13} + 7 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 2 T + p^{2} T^{2} - 46 T^{3} + 309 T^{4} - 508 T^{5} + 2499 T^{6} - 3572 T^{7} + 14537 T^{8} - 3572 p T^{9} + 2499 p^{2} T^{10} - 508 p^{3} T^{11} + 309 p^{4} T^{12} - 46 p^{5} T^{13} + p^{8} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 8 T + 76 T^{2} - 478 T^{3} + 2697 T^{4} - 13138 T^{5} + 57165 T^{6} - 218756 T^{7} + 777974 T^{8} - 218756 p T^{9} + 57165 p^{2} T^{10} - 13138 p^{3} T^{11} + 2697 p^{4} T^{12} - 478 p^{5} T^{13} + 76 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 10 T + 81 T^{2} - 392 T^{3} + 2039 T^{4} - 8668 T^{5} + 49629 T^{6} - 223908 T^{7} + 1084009 T^{8} - 223908 p T^{9} + 49629 p^{2} T^{10} - 8668 p^{3} T^{11} + 2039 p^{4} T^{12} - 392 p^{5} T^{13} + 81 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 18 T + 219 T^{2} - 1802 T^{3} + 12493 T^{4} - 72104 T^{5} + 389638 T^{6} - 1885648 T^{7} + 8689070 T^{8} - 1885648 p T^{9} + 389638 p^{2} T^{10} - 72104 p^{3} T^{11} + 12493 p^{4} T^{12} - 1802 p^{5} T^{13} + 219 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 87 T^{2} + 320 T^{3} + 4284 T^{4} + 19070 T^{5} + 146950 T^{6} + 679968 T^{7} + 3822636 T^{8} + 679968 p T^{9} + 146950 p^{2} T^{10} + 19070 p^{3} T^{11} + 4284 p^{4} T^{12} + 320 p^{5} T^{13} + 87 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 12 T + 6 p T^{2} + 1316 T^{3} + 11829 T^{4} + 71260 T^{5} + 524191 T^{6} + 2765850 T^{7} + 17529123 T^{8} + 2765850 p T^{9} + 524191 p^{2} T^{10} + 71260 p^{3} T^{11} + 11829 p^{4} T^{12} + 1316 p^{5} T^{13} + 6 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 16 T + 265 T^{2} - 2662 T^{3} + 25538 T^{4} - 190462 T^{5} + 1354956 T^{6} - 8268280 T^{7} + 48862608 T^{8} - 8268280 p T^{9} + 1354956 p^{2} T^{10} - 190462 p^{3} T^{11} + 25538 p^{4} T^{12} - 2662 p^{5} T^{13} + 265 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 24 T + 415 T^{2} - 5212 T^{3} + 55881 T^{4} - 506640 T^{5} + 4102323 T^{6} - 29242236 T^{7} + 188735749 T^{8} - 29242236 p T^{9} + 4102323 p^{2} T^{10} - 506640 p^{3} T^{11} + 55881 p^{4} T^{12} - 5212 p^{5} T^{13} + 415 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 4 T + 245 T^{2} + 1028 T^{3} + 28710 T^{4} + 115756 T^{5} + 51139 p T^{6} + 7523404 T^{7} + 103526898 T^{8} + 7523404 p T^{9} + 51139 p^{3} T^{10} + 115756 p^{3} T^{11} + 28710 p^{4} T^{12} + 1028 p^{5} T^{13} + 245 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 10 T + 229 T^{2} + 2092 T^{3} + 27208 T^{4} + 212444 T^{5} + 2061456 T^{6} + 13525358 T^{7} + 106372192 T^{8} + 13525358 p T^{9} + 2061456 p^{2} T^{10} + 212444 p^{3} T^{11} + 27208 p^{4} T^{12} + 2092 p^{5} T^{13} + 229 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 315 T^{2} - 2812 T^{3} + 45700 T^{4} - 356574 T^{5} + 3982814 T^{6} - 26481924 T^{7} + 228132200 T^{8} - 26481924 p T^{9} + 3982814 p^{2} T^{10} - 356574 p^{3} T^{11} + 45700 p^{4} T^{12} - 2812 p^{5} T^{13} + 315 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 256 T^{2} - 1200 T^{3} + 27982 T^{4} - 101214 T^{5} + 1836448 T^{6} - 5401950 T^{7} + 97827967 T^{8} - 5401950 p T^{9} + 1836448 p^{2} T^{10} - 101214 p^{3} T^{11} + 27982 p^{4} T^{12} - 1200 p^{5} T^{13} + 256 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 6 T + 365 T^{2} - 1958 T^{3} + 61846 T^{4} - 296850 T^{5} + 6442776 T^{6} - 27026362 T^{7} + 455305316 T^{8} - 27026362 p T^{9} + 6442776 p^{2} T^{10} - 296850 p^{3} T^{11} + 61846 p^{4} T^{12} - 1958 p^{5} T^{13} + 365 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 6 T + 256 T^{2} - 1160 T^{3} + 28470 T^{4} - 97782 T^{5} + 1977216 T^{6} - 5451270 T^{7} + 117546919 T^{8} - 5451270 p T^{9} + 1977216 p^{2} T^{10} - 97782 p^{3} T^{11} + 28470 p^{4} T^{12} - 1160 p^{5} T^{13} + 256 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 24 T + 605 T^{2} - 9548 T^{3} + 144468 T^{4} - 1717876 T^{5} + 19218620 T^{6} - 181264624 T^{7} + 1601805772 T^{8} - 181264624 p T^{9} + 19218620 p^{2} T^{10} - 1717876 p^{3} T^{11} + 144468 p^{4} T^{12} - 9548 p^{5} T^{13} + 605 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 42 T + 1004 T^{2} - 16258 T^{3} + 205691 T^{4} - 2191008 T^{5} + 21477229 T^{6} - 198381736 T^{7} + 1735761210 T^{8} - 198381736 p T^{9} + 21477229 p^{2} T^{10} - 2191008 p^{3} T^{11} + 205691 p^{4} T^{12} - 16258 p^{5} T^{13} + 1004 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 32 T + 773 T^{2} - 13022 T^{3} + 189699 T^{4} - 2295940 T^{5} + 25261697 T^{6} - 244556986 T^{7} + 2206859169 T^{8} - 244556986 p T^{9} + 25261697 p^{2} T^{10} - 2295940 p^{3} T^{11} + 189699 p^{4} T^{12} - 13022 p^{5} T^{13} + 773 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 2 T + 69 T^{2} + 132 T^{3} + 7640 T^{4} - 51688 T^{5} + 352216 T^{6} - 4388422 T^{7} + 25667720 T^{8} - 4388422 p T^{9} + 352216 p^{2} T^{10} - 51688 p^{3} T^{11} + 7640 p^{4} T^{12} + 132 p^{5} T^{13} + 69 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 2 T + 324 T^{2} - 248 T^{3} + 45391 T^{4} - 224408 T^{5} + 3677525 T^{6} - 40037950 T^{7} + 262136458 T^{8} - 40037950 p T^{9} + 3677525 p^{2} T^{10} - 224408 p^{3} T^{11} + 45391 p^{4} T^{12} - 248 p^{5} T^{13} + 324 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 12 T + 119 T^{2} + 424 T^{3} + 8033 T^{4} - 114408 T^{5} + 2438338 T^{6} - 4283468 T^{7} + 79370814 T^{8} - 4283468 p T^{9} + 2438338 p^{2} T^{10} - 114408 p^{3} T^{11} + 8033 p^{4} T^{12} + 424 p^{5} T^{13} + 119 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 64 T + 2281 T^{2} - 58078 T^{3} + 1164026 T^{4} - 19256938 T^{5} + 270459048 T^{6} - 3276822736 T^{7} + 34549999164 T^{8} - 3276822736 p T^{9} + 270459048 p^{2} T^{10} - 19256938 p^{3} T^{11} + 1164026 p^{4} T^{12} - 58078 p^{5} T^{13} + 2281 p^{6} T^{14} - 64 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.69132296454768227698682789996, −3.39666760637699563683650262955, −3.36997078514080371020244312358, −3.26262719605312398522197054900, −3.23737820178819374894774088378, −3.11827823976909757429644904986, −2.99800611663848346602870313114, −2.58852264145059134219126818060, −2.49257214919019554627654455739, −2.47598791969015828740755424087, −2.46393649948588858298311997294, −2.11346793546044682051860520008, −2.07307565418606907653518129469, −2.00130110601206454349971035327, −1.96844671485806211921379318932, −1.90603055566908467158630259055, −1.66227577356014225012625103072, −1.37867504615749066915191165232, −1.36644591977720030686196572540, −1.19610143385675668875036915570, −0.994318529237415926078127778524, −0.910552318615700674476454273835, −0.791283921873866277957190995171, −0.75410487873716902136650210487, −0.63729798751369223042645155824, 0.63729798751369223042645155824, 0.75410487873716902136650210487, 0.791283921873866277957190995171, 0.910552318615700674476454273835, 0.994318529237415926078127778524, 1.19610143385675668875036915570, 1.36644591977720030686196572540, 1.37867504615749066915191165232, 1.66227577356014225012625103072, 1.90603055566908467158630259055, 1.96844671485806211921379318932, 2.00130110601206454349971035327, 2.07307565418606907653518129469, 2.11346793546044682051860520008, 2.46393649948588858298311997294, 2.47598791969015828740755424087, 2.49257214919019554627654455739, 2.58852264145059134219126818060, 2.99800611663848346602870313114, 3.11827823976909757429644904986, 3.23737820178819374894774088378, 3.26262719605312398522197054900, 3.36997078514080371020244312358, 3.39666760637699563683650262955, 3.69132296454768227698682789996
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.