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: | \(8\) |
| Selberg data: | \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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
−4.11428777384891201280358001669, −4.08285059039135790600061905093, −3.70113683046593495574441751122, −3.61342624504299879899807120874, −3.59794408723174020293812047078, −3.46088471828285909106786329561, −3.36260717688569910296952425222, −3.35567479841298669818474194209, −3.03377455998282125883053200618, −2.85390847279950945675780157098, −2.80501208379741746371872214758, −2.65188534621322103997445121611, −2.61609548547692025455414834909, −2.51315785959925142912410617785, −2.47023775430054655132961683333, −2.32706642984166627661722289948, −2.25287561808061574986750889997, −2.11556989101790293111621871748, −1.73083908675437238536008686165, −1.63902782253135044482948812059, −1.50420172457986746489449359819, −1.43381897764040318744710026608, −1.18342429963103668497312566610, −1.03781454255525334606856520053, −0.855518317264551150864599140187, 0, 0, 0, 0, 0, 0, 0, 0, 0.855518317264551150864599140187, 1.03781454255525334606856520053, 1.18342429963103668497312566610, 1.43381897764040318744710026608, 1.50420172457986746489449359819, 1.63902782253135044482948812059, 1.73083908675437238536008686165, 2.11556989101790293111621871748, 2.25287561808061574986750889997, 2.32706642984166627661722289948, 2.47023775430054655132961683333, 2.51315785959925142912410617785, 2.61609548547692025455414834909, 2.65188534621322103997445121611, 2.80501208379741746371872214758, 2.85390847279950945675780157098, 3.03377455998282125883053200618, 3.35567479841298669818474194209, 3.36260717688569910296952425222, 3.46088471828285909106786329561, 3.59794408723174020293812047078, 3.61342624504299879899807120874, 3.70113683046593495574441751122, 4.08285059039135790600061905093, 4.11428777384891201280358001669
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.