Dirichlet series
| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 8·5-s − 4·6-s + 9·7-s + 9-s − 16·10-s − 11-s + 2·12-s + 13-s − 18·14-s + 16·15-s + 13·17-s − 2·18-s − 2·19-s + 8·20-s + 18·21-s + 2·22-s + 8·23-s + 36·25-s − 2·26-s + 9·28-s − 29-s − 32·30-s + 3·31-s + 2·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 3.57·5-s − 1.63·6-s + 3.40·7-s + 1/3·9-s − 5.05·10-s − 0.301·11-s + 0.577·12-s + 0.277·13-s − 4.81·14-s + 4.13·15-s + 3.15·17-s − 0.471·18-s − 0.458·19-s + 1.78·20-s + 3.92·21-s + 0.426·22-s + 1.66·23-s + 36/5·25-s − 0.392·26-s + 1.70·28-s − 0.185·29-s − 5.84·30-s + 0.538·31-s + 0.353·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.24869\times 10^{6}\) |
| Root analytic conductor: | \(2.72508\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(37.66647719\) |
| \(L(\frac12)\) | \(\approx\) | \(37.66647719\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| 5 | \( ( 1 - T )^{8} \) | |
| 31 | \( 1 - 3 T - 2 T^{2} - 171 T^{3} + 1275 T^{4} - 171 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( 1 - 9 T + 3 p T^{2} + 68 T^{3} - 459 T^{4} + 678 T^{5} + 1070 T^{6} - 4985 T^{7} + 10837 T^{8} - 4985 p T^{9} + 1070 p^{2} T^{10} + 678 p^{3} T^{11} - 459 p^{4} T^{12} + 68 p^{5} T^{13} + 3 p^{7} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + T - 12 T^{2} - 14 T^{3} + 102 T^{4} + 25 T^{5} - 1578 T^{6} - 1292 T^{7} + 21855 T^{8} - 1292 p T^{9} - 1578 p^{2} T^{10} + 25 p^{3} T^{11} + 102 p^{4} T^{12} - 14 p^{5} T^{13} - 12 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - T + 11 T^{2} + 46 T^{3} + 63 T^{4} + 38 p T^{5} + 2244 T^{6} + 3973 T^{7} + 22663 T^{8} + 3973 p T^{9} + 2244 p^{2} T^{10} + 38 p^{4} T^{11} + 63 p^{4} T^{12} + 46 p^{5} T^{13} + 11 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 13 T + 3 p T^{2} - 75 T^{3} + 741 T^{4} - 7044 T^{5} + 32220 T^{6} - 97144 T^{7} + 302007 T^{8} - 97144 p T^{9} + 32220 p^{2} T^{10} - 7044 p^{3} T^{11} + 741 p^{4} T^{12} - 75 p^{5} T^{13} + 3 p^{7} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 2 T - 33 T^{2} + 58 T^{3} + 502 T^{4} - 2300 T^{5} + 12203 T^{6} + 38654 T^{7} - 371965 T^{8} + 38654 p T^{9} + 12203 p^{2} T^{10} - 2300 p^{3} T^{11} + 502 p^{4} T^{12} + 58 p^{5} T^{13} - 33 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T - 25 T^{2} + 523 T^{3} - 1411 T^{4} - 10399 T^{5} + 70539 T^{6} + 59706 T^{7} - 1701929 T^{8} + 59706 p T^{9} + 70539 p^{2} T^{10} - 10399 p^{3} T^{11} - 1411 p^{4} T^{12} + 523 p^{5} T^{13} - 25 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + T - 64 T^{2} + 5 T^{3} + 738 T^{4} - 5240 T^{5} + 48564 T^{6} + 121646 T^{7} - 2295577 T^{8} + 121646 p T^{9} + 48564 p^{2} T^{10} - 5240 p^{3} T^{11} + 738 p^{4} T^{12} + 5 p^{5} T^{13} - 64 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 4 T + 125 T^{2} - 320 T^{3} + 6373 T^{4} - 320 p T^{5} + 125 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 29 T + 317 T^{2} - 1586 T^{3} + 5561 T^{4} - 61210 T^{5} + 692628 T^{6} - 5002185 T^{7} + 31097167 T^{8} - 5002185 p T^{9} + 692628 p^{2} T^{10} - 61210 p^{3} T^{11} + 5561 p^{4} T^{12} - 1586 p^{5} T^{13} + 317 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 25 T + 199 T^{2} - 525 T^{3} - 21223 T^{4} - 152900 T^{5} - 163378 T^{6} + 5697250 T^{7} + 55969105 T^{8} + 5697250 p T^{9} - 163378 p^{2} T^{10} - 152900 p^{3} T^{11} - 21223 p^{4} T^{12} - 525 p^{5} T^{13} + 199 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 13 T + 71 T^{2} - 455 T^{3} + 63 p T^{4} - 18124 T^{5} + 221850 T^{6} - 1925174 T^{7} + 12639677 T^{8} - 1925174 p T^{9} + 221850 p^{2} T^{10} - 18124 p^{3} T^{11} + 63 p^{5} T^{12} - 455 p^{5} T^{13} + 71 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 19 T + 89 T^{2} - 55 T^{3} + 3171 T^{4} + 45082 T^{5} + 84810 T^{6} - 532432 T^{7} - 2950573 T^{8} - 532432 p T^{9} + 84810 p^{2} T^{10} + 45082 p^{3} T^{11} + 3171 p^{4} T^{12} - 55 p^{5} T^{13} + 89 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 23 T + 252 T^{2} - 1222 T^{3} - 218 T^{4} + 39025 T^{5} - 5718 p T^{6} + 3764844 T^{7} - 36603345 T^{8} + 3764844 p T^{9} - 5718 p^{3} T^{10} + 39025 p^{3} T^{11} - 218 p^{4} T^{12} - 1222 p^{5} T^{13} + 252 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 67 | \( ( 1 + 12 T + 96 T^{2} - 3 p T^{3} - 4323 T^{4} - 3 p^{2} T^{5} + 96 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 3 T - 17 T^{2} - 309 T^{3} - 3123 T^{4} - 68400 T^{5} + 349922 T^{6} + 3062262 T^{7} + 18295685 T^{8} + 3062262 p T^{9} + 349922 p^{2} T^{10} - 68400 p^{3} T^{11} - 3123 p^{4} T^{12} - 309 p^{5} T^{13} - 17 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 14 T + 151 T^{2} + 1926 T^{3} + 21618 T^{4} + 252644 T^{5} + 2454559 T^{6} + 20660918 T^{7} + 188168803 T^{8} + 20660918 p T^{9} + 2454559 p^{2} T^{10} + 252644 p^{3} T^{11} + 21618 p^{4} T^{12} + 1926 p^{5} T^{13} + 151 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 34 T + 431 T^{2} + 1475 T^{3} - 24667 T^{4} - 320235 T^{5} - 226811 T^{6} + 30571344 T^{7} + 396344563 T^{8} + 30571344 p T^{9} - 226811 p^{2} T^{10} - 320235 p^{3} T^{11} - 24667 p^{4} T^{12} + 1475 p^{5} T^{13} + 431 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 8 T + 195 T^{2} - 1802 T^{3} + 35009 T^{4} - 242254 T^{5} + 3629289 T^{6} - 25017604 T^{7} + 349904996 T^{8} - 25017604 p T^{9} + 3629289 p^{2} T^{10} - 242254 p^{3} T^{11} + 35009 p^{4} T^{12} - 1802 p^{5} T^{13} + 195 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 25 T + 144 T^{2} + 2675 T^{3} - 56100 T^{4} + 460730 T^{5} - 462414 T^{6} - 44403550 T^{7} + 646382079 T^{8} - 44403550 p T^{9} - 462414 p^{2} T^{10} + 460730 p^{3} T^{11} - 56100 p^{4} T^{12} + 2675 p^{5} T^{13} + 144 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 28 T + 409 T^{2} + 5768 T^{3} + 78218 T^{4} + 927668 T^{5} + 10767461 T^{6} + 110281534 T^{7} + 1039547803 T^{8} + 110281534 p T^{9} + 10767461 p^{2} T^{10} + 927668 p^{3} T^{11} + 78218 p^{4} T^{12} + 5768 p^{5} T^{13} + 409 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.52107907244601068525278703509, −4.26137958303915410374450485739, −4.09813675265558195590897449624, −3.94743431622977189061495129714, −3.74584826357178766240441445600, −3.69822486091128159529057757686, −3.39695185191477711375041747135, −3.14433418389959805338643974674, −3.03168370294078469404035502030, −3.02191993563212586335758271225, −2.91018645965406846883360807628, −2.76343464223328499517372524426, −2.61263254195168879039292754331, −2.34793751716125857980694564484, −2.27498885795975199358466205900, −2.06971676085458466371922388101, −1.97550393562791628741911943565, −1.83070850298505822631364896765, −1.61298095472237521362769035246, −1.48518380780191616981374348183, −1.27018617533096463502575883950, −1.07871480637087161416812680163, −0.944985133563604803090513383712, −0.937153360723023866621436406321, −0.56441235334341701348600849001, 0.56441235334341701348600849001, 0.937153360723023866621436406321, 0.944985133563604803090513383712, 1.07871480637087161416812680163, 1.27018617533096463502575883950, 1.48518380780191616981374348183, 1.61298095472237521362769035246, 1.83070850298505822631364896765, 1.97550393562791628741911943565, 2.06971676085458466371922388101, 2.27498885795975199358466205900, 2.34793751716125857980694564484, 2.61263254195168879039292754331, 2.76343464223328499517372524426, 2.91018645965406846883360807628, 3.02191993563212586335758271225, 3.03168370294078469404035502030, 3.14433418389959805338643974674, 3.39695185191477711375041747135, 3.69822486091128159529057757686, 3.74584826357178766240441445600, 3.94743431622977189061495129714, 4.09813675265558195590897449624, 4.26137958303915410374450485739, 4.52107907244601068525278703509
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.