Dirichlet series
| L(s) = 1 | + 2·4-s − 4·7-s − 16·11-s + 16-s + 12·17-s − 24·19-s + 16·23-s + 16·25-s − 8·28-s − 24·29-s − 4·31-s + 8·37-s + 28·41-s − 8·43-s − 32·44-s + 32·47-s + 12·49-s − 16·53-s + 32·59-s − 8·61-s − 2·64-s + 24·68-s + 8·71-s − 48·76-s + 64·77-s + 81-s − 32·83-s + ⋯ |
| L(s) = 1 | + 4-s − 1.51·7-s − 4.82·11-s + 1/4·16-s + 2.91·17-s − 5.50·19-s + 3.33·23-s + 16/5·25-s − 1.51·28-s − 4.45·29-s − 0.718·31-s + 1.31·37-s + 4.37·41-s − 1.21·43-s − 4.82·44-s + 4.66·47-s + 12/7·49-s − 2.19·53-s + 4.16·59-s − 1.02·61-s − 1/4·64-s + 2.91·68-s + 0.949·71-s − 5.50·76-s + 7.29·77-s + 1/9·81-s − 3.51·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8845.73\) |
| Root analytic conductor: | \(1.76469\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.443233160\) |
| \(L(\frac12)\) | \(\approx\) | \(2.443233160\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 - T^{4} + T^{8} \) | |
| 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | |
| 13 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) | |
| good | 7 | \( 1 + 4 T + 4 T^{2} + 8 p T^{3} + 26 p T^{4} + 100 T^{5} + 32 p^{2} T^{6} + 4484 T^{7} + 1891 T^{8} + 4484 p T^{9} + 32 p^{4} T^{10} + 100 p^{3} T^{11} + 26 p^{5} T^{12} + 8 p^{6} T^{13} + 4 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 16 T + 128 T^{2} + 608 T^{3} + 1697 T^{4} + 1256 T^{5} - 12288 T^{6} - 6856 p T^{7} - 282288 T^{8} - 6856 p^{2} T^{9} - 12288 p^{2} T^{10} + 1256 p^{3} T^{11} + 1697 p^{4} T^{12} + 608 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 914 T^{4} - 588 T^{5} - 17280 T^{6} + 142356 T^{7} - 681645 T^{8} + 142356 p T^{9} - 17280 p^{2} T^{10} - 588 p^{3} T^{11} + 914 p^{4} T^{12} - 288 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 24 T + 300 T^{2} + 2496 T^{3} + 15274 T^{4} + 72600 T^{5} + 281784 T^{6} + 987048 T^{7} + 3808899 T^{8} + 987048 p T^{9} + 281784 p^{2} T^{10} + 72600 p^{3} T^{11} + 15274 p^{4} T^{12} + 2496 p^{5} T^{13} + 300 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 16 T + 98 T^{2} - 96 T^{3} - 2455 T^{4} + 15280 T^{5} - 11942 T^{6} - 360384 T^{7} + 2650708 T^{8} - 360384 p T^{9} - 11942 p^{2} T^{10} + 15280 p^{3} T^{11} - 2455 p^{4} T^{12} - 96 p^{5} T^{13} + 98 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 24 T + 320 T^{2} + 3072 T^{3} + 22497 T^{4} + 128304 T^{5} + 588544 T^{6} + 2345592 T^{7} + 10489712 T^{8} + 2345592 p T^{9} + 588544 p^{2} T^{10} + 128304 p^{3} T^{11} + 22497 p^{4} T^{12} + 3072 p^{5} T^{13} + 320 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 4 T + 8 T^{2} + 32 T^{3} + 242 T^{4} + 1172 T^{5} + 3264 T^{6} - 61188 T^{7} - 1598141 T^{8} - 61188 p T^{9} + 3264 p^{2} T^{10} + 1172 p^{3} T^{11} + 242 p^{4} T^{12} + 32 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T - 56 T^{2} + 704 T^{3} + 1625 T^{4} - 32720 T^{5} + 40232 T^{6} + 577688 T^{7} - 3274304 T^{8} + 577688 p T^{9} + 40232 p^{2} T^{10} - 32720 p^{3} T^{11} + 1625 p^{4} T^{12} + 704 p^{5} T^{13} - 56 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 28 T + 344 T^{2} - 2312 T^{3} + 7778 T^{4} + 5452 T^{5} - 220320 T^{6} + 1639100 T^{7} - 10412301 T^{8} + 1639100 p T^{9} - 220320 p^{2} T^{10} + 5452 p^{3} T^{11} + 7778 p^{4} T^{12} - 2312 p^{5} T^{13} + 344 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T + 80 T^{2} + 560 T^{3} + 2257 T^{4} + 6488 T^{5} - 25520 T^{6} - 847584 T^{7} - 4757712 T^{8} - 847584 p T^{9} - 25520 p^{2} T^{10} + 6488 p^{3} T^{11} + 2257 p^{4} T^{12} + 560 p^{5} T^{13} + 80 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( ( 1 - 16 T + 262 T^{2} - 2288 T^{3} + 19899 T^{4} - 2288 p T^{5} + 262 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 53 | \( ( 1 + 8 T + 32 T^{2} - 280 T^{3} - 5294 T^{4} - 280 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 - 32 T + 464 T^{2} - 3024 T^{3} - 5311 T^{4} + 233864 T^{5} - 834224 T^{6} - 16269912 T^{7} + 222050128 T^{8} - 16269912 p T^{9} - 834224 p^{2} T^{10} + 233864 p^{3} T^{11} - 5311 p^{4} T^{12} - 3024 p^{5} T^{13} + 464 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 8 T - 56 T^{2} - 1616 T^{3} - 3586 T^{4} + 110216 T^{5} + 1001600 T^{6} - 3573896 T^{7} - 80557133 T^{8} - 3573896 p T^{9} + 1001600 p^{2} T^{10} + 110216 p^{3} T^{11} - 3586 p^{4} T^{12} - 1616 p^{5} T^{13} - 56 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 200 T^{2} + 22174 T^{4} + 1769600 T^{6} + 118055155 T^{8} + 1769600 p^{2} T^{10} + 22174 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 - 8 T + 128 T^{2} - 1888 T^{3} + 21950 T^{4} - 211576 T^{5} + 2372928 T^{6} - 20475800 T^{7} + 185905251 T^{8} - 20475800 p T^{9} + 2372928 p^{2} T^{10} - 211576 p^{3} T^{11} + 21950 p^{4} T^{12} - 1888 p^{5} T^{13} + 128 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 248 T^{2} + 40260 T^{4} - 4471720 T^{6} + 376225286 T^{8} - 4471720 p^{2} T^{10} + 40260 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 - 308 T^{2} + 55642 T^{4} - 6901040 T^{6} + 629487139 T^{8} - 6901040 p^{2} T^{10} + 55642 p^{4} T^{12} - 308 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 16 T + 184 T^{2} + 368 T^{3} + 1314 T^{4} + 368 p T^{5} + 184 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 4 T + 200 T^{2} - 256 T^{3} + 8738 T^{4} - 89068 T^{5} - 79680 T^{6} + 1434052 T^{7} + 8784243 T^{8} + 1434052 p T^{9} - 79680 p^{2} T^{10} - 89068 p^{3} T^{11} + 8738 p^{4} T^{12} - 256 p^{5} T^{13} + 200 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 12 T + 404 T^{2} - 4272 T^{3} + 89014 T^{4} - 836052 T^{5} + 13547360 T^{6} - 1157724 p T^{7} + 15794947 p T^{8} - 1157724 p^{2} T^{9} + 13547360 p^{2} T^{10} - 836052 p^{3} T^{11} + 89014 p^{4} T^{12} - 4272 p^{5} T^{13} + 404 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.17739428814890195762831522638, −4.89950140843960021266587718054, −4.74242394776852288219076691041, −4.55036944826653502975498713200, −4.46046809530004339248908941660, −4.37508987666113162499529834501, −3.99907795732295306082684097310, −3.98016911780624111293283810285, −3.97701724763419902481156047237, −3.58857992035492768200194475695, −3.44733099803808610025648068257, −3.17547184407504068945141532815, −3.05059978159542802998633991509, −2.96443353568233532355298971802, −2.68935401759882476391343039123, −2.68552850555238078552441781907, −2.52147052455716793953828149440, −2.42246699272583060347722261145, −2.17617346653506787990822546636, −2.06075250906399650600343360328, −1.74687074324416179597910832543, −1.48181908890202348556386294835, −0.67958252510744377457143937226, −0.64710594996441941816302525624, −0.55886571739866046879485687441, 0.55886571739866046879485687441, 0.64710594996441941816302525624, 0.67958252510744377457143937226, 1.48181908890202348556386294835, 1.74687074324416179597910832543, 2.06075250906399650600343360328, 2.17617346653506787990822546636, 2.42246699272583060347722261145, 2.52147052455716793953828149440, 2.68552850555238078552441781907, 2.68935401759882476391343039123, 2.96443353568233532355298971802, 3.05059978159542802998633991509, 3.17547184407504068945141532815, 3.44733099803808610025648068257, 3.58857992035492768200194475695, 3.97701724763419902481156047237, 3.98016911780624111293283810285, 3.99907795732295306082684097310, 4.37508987666113162499529834501, 4.46046809530004339248908941660, 4.55036944826653502975498713200, 4.74242394776852288219076691041, 4.89950140843960021266587718054, 5.17739428814890195762831522638
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.