Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 4·5-s + 8·6-s + 2·7-s + 11·9-s + 8·10-s − 2·11-s − 4·12-s − 11·13-s − 4·14-s + 16·15-s − 7·17-s − 22·18-s − 14·19-s − 4·20-s − 8·21-s + 4·22-s + 3·23-s − 12·25-s + 22·26-s − 22·27-s + 2·28-s + 13·29-s − 32·30-s + 15·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s + 0.755·7-s + 11/3·9-s + 2.52·10-s − 0.603·11-s − 1.15·12-s − 3.05·13-s − 1.06·14-s + 4.13·15-s − 1.69·17-s − 5.18·18-s − 3.21·19-s − 0.894·20-s − 1.74·21-s + 0.852·22-s + 0.625·23-s − 2.39·25-s + 4.31·26-s − 4.23·27-s + 0.377·28-s + 2.41·29-s − 5.84·30-s + 2.69·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00360869\) |
| Root analytic conductor: | \(0.703613\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03052412083\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03052412083\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 - 15 T + 136 T^{2} - 855 T^{3} + 5011 T^{4} - 855 p T^{5} + 136 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + 4 T + 5 T^{2} - 2 T^{3} - 4 T^{4} + 46 T^{5} + 109 T^{6} - 221 T^{8} + 109 p^{2} T^{10} + 46 p^{3} T^{11} - 4 p^{4} T^{12} - 2 p^{5} T^{13} + 5 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 2 T + 12 T^{2} + 21 T^{3} + 79 T^{4} + 21 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 7 | \( 1 - 2 T - 6 T^{3} + T^{4} - 218 T^{5} + 536 T^{6} + 160 T^{7} + 449 T^{8} + 160 p T^{9} + 536 p^{2} T^{10} - 218 p^{3} T^{11} + p^{4} T^{12} - 6 p^{5} T^{13} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T - 13 T^{2} - 93 T^{3} - 67 T^{4} + 1217 T^{5} + 3765 T^{6} - 5340 T^{7} - 57089 T^{8} - 5340 p T^{9} + 3765 p^{2} T^{10} + 1217 p^{3} T^{11} - 67 p^{4} T^{12} - 93 p^{5} T^{13} - 13 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 11 T + 42 T^{2} + 51 T^{3} - 48 T^{4} - 606 T^{5} - 4614 T^{6} - 12918 T^{7} - 18633 T^{8} - 12918 p T^{9} - 4614 p^{2} T^{10} - 606 p^{3} T^{11} - 48 p^{4} T^{12} + 51 p^{5} T^{13} + 42 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 7 T - 27 T^{2} - 322 T^{3} - 223 T^{4} + 7078 T^{5} + 31324 T^{6} - 54369 T^{7} - 777193 T^{8} - 54369 p T^{9} + 31324 p^{2} T^{10} + 7078 p^{3} T^{11} - 223 p^{4} T^{12} - 322 p^{5} T^{13} - 27 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 14 T + 113 T^{2} + 791 T^{3} + 5193 T^{4} + 29519 T^{5} + 149985 T^{6} + 739620 T^{7} + 3415831 T^{8} + 739620 p T^{9} + 149985 p^{2} T^{10} + 29519 p^{3} T^{11} + 5193 p^{4} T^{12} + 791 p^{5} T^{13} + 113 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 3 T - 24 T^{2} + 9 p T^{3} + 214 T^{4} - 942 T^{5} - 5478 T^{6} - 4254 T^{7} + 582073 T^{8} - 4254 p T^{9} - 5478 p^{2} T^{10} - 942 p^{3} T^{11} + 214 p^{4} T^{12} + 9 p^{6} T^{13} - 24 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 13 T + 51 T^{2} - 286 T^{3} + 3965 T^{4} - 25648 T^{5} + 112840 T^{6} - 715395 T^{7} + 4605311 T^{8} - 715395 p T^{9} + 112840 p^{2} T^{10} - 25648 p^{3} T^{11} + 3965 p^{4} T^{12} - 286 p^{5} T^{13} + 51 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 26 T + 382 T^{2} - 3761 T^{3} + 26675 T^{4} - 3761 p T^{5} + 382 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 11 T + 51 T^{2} + 2 p T^{3} + 197 T^{4} - 1624 T^{5} - 928 p T^{6} - 12993 p T^{7} - 3785809 T^{8} - 12993 p^{2} T^{9} - 928 p^{3} T^{10} - 1624 p^{3} T^{11} + 197 p^{4} T^{12} + 2 p^{6} T^{13} + 51 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 22 T + 216 T^{2} + 1402 T^{3} + 7969 T^{4} + 16318 T^{5} - 336768 T^{6} - 4240984 T^{7} - 30496247 T^{8} - 4240984 p T^{9} - 336768 p^{2} T^{10} + 16318 p^{3} T^{11} + 7969 p^{4} T^{12} + 1402 p^{5} T^{13} + 216 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 10 T + 250 T^{3} + 4721 T^{4} + 17410 T^{5} + 220960 T^{6} + 1700040 T^{7} + 4996121 T^{8} + 1700040 p T^{9} + 220960 p^{2} T^{10} + 17410 p^{3} T^{11} + 4721 p^{4} T^{12} + 250 p^{5} T^{13} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 7 T + 2 T^{2} - 36 T^{3} + 4286 T^{4} + 2813 T^{5} - 253200 T^{6} + 1144962 T^{7} + 5274439 T^{8} + 1144962 p T^{9} - 253200 p^{2} T^{10} + 2813 p^{3} T^{11} + 4286 p^{4} T^{12} - 36 p^{5} T^{13} + 2 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 22 T + 261 T^{2} - 2224 T^{3} + 16610 T^{4} - 41842 T^{5} - 752825 T^{6} + 11709210 T^{7} - 97367449 T^{8} + 11709210 p T^{9} - 752825 p^{2} T^{10} - 41842 p^{3} T^{11} + 16610 p^{4} T^{12} - 2224 p^{5} T^{13} + 261 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 - 2 T + 156 T^{2} - 252 T^{3} + 11869 T^{4} - 252 p T^{5} + 156 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 13 T + 292 T^{2} + 2485 T^{3} + 30011 T^{4} + 2485 p T^{5} + 292 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 15 T + 92 T^{2} + 420 T^{3} + 7158 T^{4} - 1875 T^{5} - 853196 T^{6} - 8594910 T^{7} - 46748785 T^{8} - 8594910 p T^{9} - 853196 p^{2} T^{10} - 1875 p^{3} T^{11} + 7158 p^{4} T^{12} + 420 p^{5} T^{13} + 92 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 29 T + 360 T^{2} + 2253 T^{3} + 8406 T^{4} + 49896 T^{5} - 185556 T^{6} - 16355970 T^{7} - 217547421 T^{8} - 16355970 p T^{9} - 185556 p^{2} T^{10} + 49896 p^{3} T^{11} + 8406 p^{4} T^{12} + 2253 p^{5} T^{13} + 360 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 2 T - 69 T^{2} + 549 T^{3} - 3015 T^{4} - 142593 T^{5} + 703845 T^{6} + 6633780 T^{7} - 63727749 T^{8} + 6633780 p T^{9} + 703845 p^{2} T^{10} - 142593 p^{3} T^{11} - 3015 p^{4} T^{12} + 549 p^{5} T^{13} - 69 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 38 T + 816 T^{2} - 13718 T^{3} + 200009 T^{4} - 2575982 T^{5} + 29683912 T^{6} - 311229264 T^{7} + 2977020113 T^{8} - 311229264 p T^{9} + 29683912 p^{2} T^{10} - 2575982 p^{3} T^{11} + 200009 p^{4} T^{12} - 13718 p^{5} T^{13} + 816 p^{6} T^{14} - 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - T + 108 T^{2} - 824 T^{3} + 13658 T^{4} - 11251 T^{5} + 1304650 T^{6} - 440970 T^{7} + 85530491 T^{8} - 440970 p T^{9} + 1304650 p^{2} T^{10} - 11251 p^{3} T^{11} + 13658 p^{4} T^{12} - 824 p^{5} T^{13} + 108 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 10 T + 180 T^{2} + 1570 T^{3} + 27781 T^{4} + 369370 T^{5} + 3658200 T^{6} + 43878740 T^{7} + 329787001 T^{8} + 43878740 p T^{9} + 3658200 p^{2} T^{10} + 369370 p^{3} T^{11} + 27781 p^{4} T^{12} + 1570 p^{5} T^{13} + 180 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42762993573266940455642050829, −7.02433093754683480321980441038, −6.92411157106339607658690577727, −6.70905901732532507096933884127, −6.67326917599417497162834425540, −6.32879942218616763432580163970, −6.28958323912899361077644733054, −6.06342779638369951582640801206, −5.98649536169807888639759264442, −5.67973287564568137400141997920, −5.31825142155208792642946489868, −5.17421994580105140924874448250, −4.68812954894910771822191929594, −4.62471239985438708758071791465, −4.45900780904558648544305623658, −4.38077387267054957572429698924, −4.35913510148740032391222036256, −4.33499688441599173565019927675, −3.90876293882549813645689938395, −3.14034038110398688374794328790, −2.72568073924399445212118744854, −2.72515917977716506852677048302, −2.16735905806938074598772021945, −1.93691816174177389332914676990, −0.52794830710636374841122311341, 0.52794830710636374841122311341, 1.93691816174177389332914676990, 2.16735905806938074598772021945, 2.72515917977716506852677048302, 2.72568073924399445212118744854, 3.14034038110398688374794328790, 3.90876293882549813645689938395, 4.33499688441599173565019927675, 4.35913510148740032391222036256, 4.38077387267054957572429698924, 4.45900780904558648544305623658, 4.62471239985438708758071791465, 4.68812954894910771822191929594, 5.17421994580105140924874448250, 5.31825142155208792642946489868, 5.67973287564568137400141997920, 5.98649536169807888639759264442, 6.06342779638369951582640801206, 6.28958323912899361077644733054, 6.32879942218616763432580163970, 6.67326917599417497162834425540, 6.70905901732532507096933884127, 6.92411157106339607658690577727, 7.02433093754683480321980441038, 7.42762993573266940455642050829
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.