Dirichlet series
| L(s) = 1 | − 2-s + 6·3-s − 4-s + 3·5-s − 6·6-s + 2·7-s − 2·8-s + 25·9-s − 3·10-s − 6·12-s − 5·13-s − 2·14-s + 18·15-s + 4·16-s + 14·17-s − 25·18-s + 6·19-s − 3·20-s + 12·21-s − 16·23-s − 12·24-s + 7·25-s + 5·26-s + 88·27-s − 2·28-s + 6·29-s − 18·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.46·3-s − 1/2·4-s + 1.34·5-s − 2.44·6-s + 0.755·7-s − 0.707·8-s + 25/3·9-s − 0.948·10-s − 1.73·12-s − 1.38·13-s − 0.534·14-s + 4.64·15-s + 16-s + 3.39·17-s − 5.89·18-s + 1.37·19-s − 0.670·20-s + 2.61·21-s − 3.33·23-s − 2.44·24-s + 7/5·25-s + 0.980·26-s + 16.9·27-s − 0.377·28-s + 1.11·29-s − 3.28·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.37808\times 10^{6}\) |
| Root analytic conductor: | \(2.60064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(55.81028738\) |
| \(L(\frac12)\) | \(\approx\) | \(55.81028738\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} + 5 T^{3} + 5 T^{4} + 19 T^{5} + 13 p T^{6} + 25 T^{7} + 59 T^{8} + 25 p T^{9} + 13 p^{3} T^{10} + 19 p^{3} T^{11} + 5 p^{4} T^{12} + 5 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 - p T + T^{2} + T^{3} + 4 T^{4} + p T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \) | |
| 5 | \( 1 - 3 T + 2 T^{2} + 6 T^{4} - 3 p^{2} T^{5} + 208 T^{6} - 234 T^{7} + 431 T^{8} - 234 p T^{9} + 208 p^{2} T^{10} - 3 p^{5} T^{11} + 6 p^{4} T^{12} + 2 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 5 T - 30 T^{2} - 170 T^{3} + 22 p T^{4} + 2815 T^{5} + 5030 T^{6} - 15820 T^{7} - 130849 T^{8} - 15820 p T^{9} + 5030 p^{2} T^{10} + 2815 p^{3} T^{11} + 22 p^{5} T^{12} - 170 p^{5} T^{13} - 30 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 14 T + 79 T^{2} - 281 T^{3} + 1089 T^{4} - 2031 T^{5} - 15507 T^{6} + 106442 T^{7} - 393507 T^{8} + 106442 p T^{9} - 15507 p^{2} T^{10} - 2031 p^{3} T^{11} + 1089 p^{4} T^{12} - 281 p^{5} T^{13} + 79 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 6 T - 27 T^{2} + 111 T^{3} + 483 T^{4} - 2061 T^{5} + 20515 T^{6} - 13260 T^{7} - 552129 T^{8} - 13260 p T^{9} + 20515 p^{2} T^{10} - 2061 p^{3} T^{11} + 483 p^{4} T^{12} + 111 p^{5} T^{13} - 27 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 402 p T^{5} + 83 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 6 T + 23 T^{2} + 6 T^{3} + 138 T^{4} + 5484 T^{5} - 19805 T^{6} + 121230 T^{7} + 57131 T^{8} + 121230 p T^{9} - 19805 p^{2} T^{10} + 5484 p^{3} T^{11} + 138 p^{4} T^{12} + 6 p^{5} T^{13} + 23 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 14 T + 29 T^{2} + 8 p T^{3} + 530 T^{4} - 15954 T^{5} + 65935 T^{6} - 166230 T^{7} + 667951 T^{8} - 166230 p T^{9} + 65935 p^{2} T^{10} - 15954 p^{3} T^{11} + 530 p^{4} T^{12} + 8 p^{6} T^{13} + 29 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T - 48 T^{2} + 175 T^{3} + 2760 T^{4} - 1334 T^{5} - 138534 T^{6} - 40270 T^{7} + 6227939 T^{8} - 40270 p T^{9} - 138534 p^{2} T^{10} - 1334 p^{3} T^{11} + 2760 p^{4} T^{12} + 175 p^{5} T^{13} - 48 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 18 T + 145 T^{2} - 1046 T^{3} + 9461 T^{4} - 59718 T^{5} + 237483 T^{6} - 1463634 T^{7} + 11961308 T^{8} - 1463634 p T^{9} + 237483 p^{2} T^{10} - 59718 p^{3} T^{11} + 9461 p^{4} T^{12} - 1046 p^{5} T^{13} + 145 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 47 | \( 1 - 7 T + 85 T^{2} - 1176 T^{3} + 9351 T^{4} - 81508 T^{5} + 731706 T^{6} - 5276705 T^{7} + 35329489 T^{8} - 5276705 p T^{9} + 731706 p^{2} T^{10} - 81508 p^{3} T^{11} + 9351 p^{4} T^{12} - 1176 p^{5} T^{13} + 85 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 7 T + 132 T^{2} - 1421 T^{3} + 16876 T^{4} - 131512 T^{5} + 1370850 T^{6} - 9985608 T^{7} + 80785479 T^{8} - 9985608 p T^{9} + 1370850 p^{2} T^{10} - 131512 p^{3} T^{11} + 16876 p^{4} T^{12} - 1421 p^{5} T^{13} + 132 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 9 T^{2} + 800 T^{3} + 2280 T^{4} - 15200 T^{5} + 247289 T^{6} + 987600 T^{7} - 4469641 T^{8} + 987600 p T^{9} + 247289 p^{2} T^{10} - 15200 p^{3} T^{11} + 2280 p^{4} T^{12} + 800 p^{5} T^{13} - 9 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 - 12 T + 147 T^{2} - 2312 T^{3} + 22368 T^{4} - 224532 T^{5} + 2244305 T^{6} - 18537880 T^{7} + 147792991 T^{8} - 18537880 p T^{9} + 2244305 p^{2} T^{10} - 224532 p^{3} T^{11} + 22368 p^{4} T^{12} - 2312 p^{5} T^{13} + 147 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 21 T + 151 T^{2} - 7 p T^{3} + 5457 T^{4} - 14896 T^{5} - 888488 T^{6} + 11215568 T^{7} - 84203719 T^{8} + 11215568 p T^{9} - 888488 p^{2} T^{10} - 14896 p^{3} T^{11} + 5457 p^{4} T^{12} - 7 p^{6} T^{13} + 151 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 8 T - 77 T^{2} + 990 T^{3} - 2170 T^{4} - 132362 T^{5} + 1139369 T^{6} + 4655830 T^{7} - 91523921 T^{8} + 4655830 p T^{9} + 1139369 p^{2} T^{10} - 132362 p^{3} T^{11} - 2170 p^{4} T^{12} + 990 p^{5} T^{13} - 77 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - T - 160 T^{2} - 548 T^{3} + 13806 T^{4} + 31769 T^{5} - 282588 T^{6} - 2159462 T^{7} + 16750183 T^{8} - 2159462 p T^{9} - 282588 p^{2} T^{10} + 31769 p^{3} T^{11} + 13806 p^{4} T^{12} - 548 p^{5} T^{13} - 160 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 22 T + 313 T^{2} + 2340 T^{3} + 7050 T^{4} - 116402 T^{5} - 1950441 T^{6} - 18988930 T^{7} - 158760641 T^{8} - 18988930 p T^{9} - 1950441 p^{2} T^{10} - 116402 p^{3} T^{11} + 7050 p^{4} T^{12} + 2340 p^{5} T^{13} + 313 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 3419 p T^{5} + 312 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 15 T + 36 T^{2} + 375 T^{3} + 2412 T^{4} - 52890 T^{5} + 1250458 T^{6} + 18589350 T^{7} + 83790255 T^{8} + 18589350 p T^{9} + 1250458 p^{2} T^{10} - 52890 p^{3} T^{11} + 2412 p^{4} T^{12} + 375 p^{5} T^{13} + 36 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.18758916412974002628084998388, −4.12203939360168752339747225631, −4.11476587604019711527791699008, −4.03485183974874501297179507899, −3.93664531565468987904684266231, −3.75456235363800616077346849307, −3.65784665133483042670310846833, −3.14685542323734084882103498293, −3.09620282354486687999353903993, −3.03883509324372908501173119260, −2.92026778860601825466831512335, −2.83703192614371932549059205446, −2.83115382758914633017319406974, −2.79918026244723281152199411402, −2.34927234535923147752009460123, −2.22003102064597639440761195963, −1.94641144038456692569063432557, −1.89144300016251709289998031199, −1.88804626981884752605416382560, −1.70870658201189320930174432050, −1.24450374876580014803841359465, −1.14399102676611758600818929582, −0.816100469152108448182357842755, −0.814504688715402890006025281401, −0.72503681648671487599706827213, 0.72503681648671487599706827213, 0.814504688715402890006025281401, 0.816100469152108448182357842755, 1.14399102676611758600818929582, 1.24450374876580014803841359465, 1.70870658201189320930174432050, 1.88804626981884752605416382560, 1.89144300016251709289998031199, 1.94641144038456692569063432557, 2.22003102064597639440761195963, 2.34927234535923147752009460123, 2.79918026244723281152199411402, 2.83115382758914633017319406974, 2.83703192614371932549059205446, 2.92026778860601825466831512335, 3.03883509324372908501173119260, 3.09620282354486687999353903993, 3.14685542323734084882103498293, 3.65784665133483042670310846833, 3.75456235363800616077346849307, 3.93664531565468987904684266231, 4.03485183974874501297179507899, 4.11476587604019711527791699008, 4.12203939360168752339747225631, 4.18758916412974002628084998388
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.