Dirichlet series
| L(s) = 1 | + 4·3-s + 4·5-s − 4·11-s + 20·13-s + 16·15-s − 8·17-s + 8·19-s + 4·23-s − 8·25-s − 16·27-s − 16·29-s − 8·31-s − 16·33-s + 8·37-s + 80·39-s + 12·41-s − 4·43-s + 4·47-s − 32·51-s − 16·55-s + 32·57-s − 16·59-s + 32·61-s + 80·65-s + 16·69-s + 24·73-s − 32·75-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s − 1.20·11-s + 5.54·13-s + 4.13·15-s − 1.94·17-s + 1.83·19-s + 0.834·23-s − 8/5·25-s − 3.07·27-s − 2.97·29-s − 1.43·31-s − 2.78·33-s + 1.31·37-s + 12.8·39-s + 1.87·41-s − 0.609·43-s + 0.583·47-s − 4.48·51-s − 2.15·55-s + 4.23·57-s − 2.08·59-s + 4.09·61-s + 9.92·65-s + 1.92·69-s + 2.80·73-s − 3.69·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 7^{16} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.51105\times 10^{11}\) |
| Root analytic conductor: | \(5.15811\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(96.89492959\) |
| \(L(\frac12)\) | \(\approx\) | \(96.89492959\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 17 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 - 4 T + 16 T^{2} - 16 p T^{3} + 125 T^{4} - 292 T^{5} + 616 T^{6} - 1180 T^{7} + 2165 T^{8} - 1180 p T^{9} + 616 p^{2} T^{10} - 292 p^{3} T^{11} + 125 p^{4} T^{12} - 16 p^{6} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 4 T + 24 T^{2} - 84 T^{3} + 61 p T^{4} - 172 p T^{5} + 496 p T^{6} - 5972 T^{7} + 14293 T^{8} - 5972 p T^{9} + 496 p^{3} T^{10} - 172 p^{4} T^{11} + 61 p^{5} T^{12} - 84 p^{5} T^{13} + 24 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 4 T + 36 T^{2} + 156 T^{3} + 772 T^{4} + 260 p T^{5} + 11940 T^{6} + 39604 T^{7} + 142038 T^{8} + 39604 p T^{9} + 11940 p^{2} T^{10} + 260 p^{4} T^{11} + 772 p^{4} T^{12} + 156 p^{5} T^{13} + 36 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 20 T + 242 T^{2} - 2108 T^{3} + 14688 T^{4} - 85220 T^{5} + 425054 T^{6} - 1848812 T^{7} + 7093070 T^{8} - 1848812 p T^{9} + 425054 p^{2} T^{10} - 85220 p^{3} T^{11} + 14688 p^{4} T^{12} - 2108 p^{5} T^{13} + 242 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 8 T + 130 T^{2} - 768 T^{3} + 7300 T^{4} - 34608 T^{5} + 246198 T^{6} - 51080 p T^{7} + 5606838 T^{8} - 51080 p^{2} T^{9} + 246198 p^{2} T^{10} - 34608 p^{3} T^{11} + 7300 p^{4} T^{12} - 768 p^{5} T^{13} + 130 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 4 T + 90 T^{2} - 540 T^{3} + 4440 T^{4} - 29276 T^{5} + 164982 T^{6} - 942116 T^{7} + 4535198 T^{8} - 942116 p T^{9} + 164982 p^{2} T^{10} - 29276 p^{3} T^{11} + 4440 p^{4} T^{12} - 540 p^{5} T^{13} + 90 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 16 T + 8 p T^{2} + 2288 T^{3} + 20820 T^{4} + 156880 T^{5} + 1107792 T^{6} + 6773040 T^{7} + 39031606 T^{8} + 6773040 p T^{9} + 1107792 p^{2} T^{10} + 156880 p^{3} T^{11} + 20820 p^{4} T^{12} + 2288 p^{5} T^{13} + 8 p^{7} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 8 T + 132 T^{2} + 1148 T^{3} + 339 p T^{4} + 76248 T^{5} + 557672 T^{6} + 3391732 T^{7} + 20312877 T^{8} + 3391732 p T^{9} + 557672 p^{2} T^{10} + 76248 p^{3} T^{11} + 339 p^{5} T^{12} + 1148 p^{5} T^{13} + 132 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 262 T^{2} - 1752 T^{3} + 30896 T^{4} - 174584 T^{5} + 2158362 T^{6} - 10209672 T^{7} + 97900174 T^{8} - 10209672 p T^{9} + 2158362 p^{2} T^{10} - 174584 p^{3} T^{11} + 30896 p^{4} T^{12} - 1752 p^{5} T^{13} + 262 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 12 T + 252 T^{2} - 2556 T^{3} + 31145 T^{4} - 259212 T^{5} + 2352656 T^{6} - 16138476 T^{7} + 117350157 T^{8} - 16138476 p T^{9} + 2352656 p^{2} T^{10} - 259212 p^{3} T^{11} + 31145 p^{4} T^{12} - 2556 p^{5} T^{13} + 252 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 4 T + 134 T^{2} - 32 T^{3} + 6463 T^{4} - 33524 T^{5} + 283988 T^{6} - 1881912 T^{7} + 14761517 T^{8} - 1881912 p T^{9} + 283988 p^{2} T^{10} - 33524 p^{3} T^{11} + 6463 p^{4} T^{12} - 32 p^{5} T^{13} + 134 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 4 T + 212 T^{2} - 620 T^{3} + 19692 T^{4} - 29500 T^{5} + 1107892 T^{6} - 256724 T^{7} + 51399558 T^{8} - 256724 p T^{9} + 1107892 p^{2} T^{10} - 29500 p^{3} T^{11} + 19692 p^{4} T^{12} - 620 p^{5} T^{13} + 212 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 182 T^{2} + 396 T^{3} + 18127 T^{4} + 76688 T^{5} + 1202396 T^{6} + 7128472 T^{7} + 67495349 T^{8} + 7128472 p T^{9} + 1202396 p^{2} T^{10} + 76688 p^{3} T^{11} + 18127 p^{4} T^{12} + 396 p^{5} T^{13} + 182 p^{6} T^{14} + p^{8} T^{16} \) | |
| 59 | \( 1 + 16 T + 308 T^{2} + 3024 T^{3} + 35584 T^{4} + 259920 T^{5} + 2411692 T^{6} + 14703504 T^{7} + 135177774 T^{8} + 14703504 p T^{9} + 2411692 p^{2} T^{10} + 259920 p^{3} T^{11} + 35584 p^{4} T^{12} + 3024 p^{5} T^{13} + 308 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 32 T + 780 T^{2} - 13760 T^{3} + 202241 T^{4} - 2496952 T^{5} + 26836056 T^{6} - 251837332 T^{7} + 2095384045 T^{8} - 251837332 p T^{9} + 26836056 p^{2} T^{10} - 2496952 p^{3} T^{11} + 202241 p^{4} T^{12} - 13760 p^{5} T^{13} + 780 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 182 T^{2} - 380 T^{3} + 20055 T^{4} - 58520 T^{5} + 1944308 T^{6} - 4410912 T^{7} + 151707053 T^{8} - 4410912 p T^{9} + 1944308 p^{2} T^{10} - 58520 p^{3} T^{11} + 20055 p^{4} T^{12} - 380 p^{5} T^{13} + 182 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 + 282 T^{2} - 120 T^{3} + 39300 T^{4} + 5592 T^{5} + 3710766 T^{6} + 2456208 T^{7} + 279763670 T^{8} + 2456208 p T^{9} + 3710766 p^{2} T^{10} + 5592 p^{3} T^{11} + 39300 p^{4} T^{12} - 120 p^{5} T^{13} + 282 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 - 24 T + 532 T^{2} - 7840 T^{3} + 111593 T^{4} - 1287208 T^{5} + 14379232 T^{6} - 136783428 T^{7} + 1257242741 T^{8} - 136783428 p T^{9} + 14379232 p^{2} T^{10} - 1287208 p^{3} T^{11} + 111593 p^{4} T^{12} - 7840 p^{5} T^{13} + 532 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 4 T + 412 T^{2} - 980 T^{3} + 78884 T^{4} - 128196 T^{5} + 9972860 T^{6} - 14036628 T^{7} + 919676534 T^{8} - 14036628 p T^{9} + 9972860 p^{2} T^{10} - 128196 p^{3} T^{11} + 78884 p^{4} T^{12} - 980 p^{5} T^{13} + 412 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 28 T + 590 T^{2} - 8124 T^{3} + 100500 T^{4} - 988060 T^{5} + 9343514 T^{6} - 75809692 T^{7} + 692605078 T^{8} - 75809692 p T^{9} + 9343514 p^{2} T^{10} - 988060 p^{3} T^{11} + 100500 p^{4} T^{12} - 8124 p^{5} T^{13} + 590 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 20 T + 628 T^{2} - 9348 T^{3} + 170452 T^{4} - 2031132 T^{5} + 27607732 T^{6} - 271275276 T^{7} + 2973835350 T^{8} - 271275276 p T^{9} + 27607732 p^{2} T^{10} - 2031132 p^{3} T^{11} + 170452 p^{4} T^{12} - 9348 p^{5} T^{13} + 628 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 56 T + 2040 T^{2} - 52648 T^{3} + 1089417 T^{4} - 18429928 T^{5} + 264789328 T^{6} - 3243680020 T^{7} + 34414745117 T^{8} - 3243680020 p T^{9} + 264789328 p^{2} T^{10} - 18429928 p^{3} T^{11} + 1089417 p^{4} T^{12} - 52648 p^{5} T^{13} + 2040 p^{6} T^{14} - 56 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.66815107272191255448819501807, −3.36722640436760921509794672440, −3.31074229452828336802700565226, −3.19908917824545086579352607243, −3.10503167744979575370749266539, −2.96466516312116932419056124281, −2.96364176519670467990755345573, −2.77501668657397723823843904395, −2.73951632506840970770739499870, −2.27741904580852919683964362800, −2.22998758619064200829577280762, −2.21431248410130303331426350607, −2.12185972904597207072534096091, −2.05236014544039079387587479111, −1.95318806206666422542220214243, −1.88753429925440673377817815430, −1.86967287483556514081064458890, −1.38817457350297941992360574867, −1.37580546528844004142723143907, −1.17040288994534890482874969269, −0.885259736099810619523120277286, −0.794758986820798593238873719151, −0.61832347012825404223812651152, −0.46589661966613965993927666612, −0.43475734108515463502525547193, 0.43475734108515463502525547193, 0.46589661966613965993927666612, 0.61832347012825404223812651152, 0.794758986820798593238873719151, 0.885259736099810619523120277286, 1.17040288994534890482874969269, 1.37580546528844004142723143907, 1.38817457350297941992360574867, 1.86967287483556514081064458890, 1.88753429925440673377817815430, 1.95318806206666422542220214243, 2.05236014544039079387587479111, 2.12185972904597207072534096091, 2.21431248410130303331426350607, 2.22998758619064200829577280762, 2.27741904580852919683964362800, 2.73951632506840970770739499870, 2.77501668657397723823843904395, 2.96364176519670467990755345573, 2.96466516312116932419056124281, 3.10503167744979575370749266539, 3.19908917824545086579352607243, 3.31074229452828336802700565226, 3.36722640436760921509794672440, 3.66815107272191255448819501807
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.