Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s + 2·4-s − 4·5-s + 3·6-s − 8·7-s + 5·8-s + 10·9-s + 4·10-s − 4·11-s − 6·12-s − 7·13-s + 8·14-s + 12·15-s − 4·16-s + 17-s − 10·18-s + 5·19-s − 8·20-s + 24·21-s + 4·22-s − 2·23-s − 15·24-s + 6·25-s + 7·26-s − 19·27-s − 16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 4-s − 1.78·5-s + 1.22·6-s − 3.02·7-s + 1.76·8-s + 10/3·9-s + 1.26·10-s − 1.20·11-s − 1.73·12-s − 1.94·13-s + 2.13·14-s + 3.09·15-s − 16-s + 0.242·17-s − 2.35·18-s + 1.14·19-s − 1.78·20-s + 5.23·21-s + 0.852·22-s − 0.417·23-s − 3.06·24-s + 6/5·25-s + 1.37·26-s − 3.65·27-s − 3.02·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.109649\) |
| Root analytic conductor: | \(0.870964\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1542590080\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1542590080\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | \( 1 - 5 T + 31 T^{2} - 67 T^{3} + 395 T^{4} - 67 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 + T - T^{2} - p^{3} T^{3} - 7 T^{4} + p^{3} T^{5} + 19 T^{6} + 3 p T^{7} - 15 p T^{8} + 3 p^{2} T^{9} + 19 p^{2} T^{10} + p^{6} T^{11} - 7 p^{4} T^{12} - p^{8} T^{13} - p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + p T - T^{2} - 14 T^{3} - 10 T^{4} + 22 T^{5} + 26 T^{6} - T^{7} - 11 T^{8} - p T^{9} + 26 p^{2} T^{10} + 22 p^{3} T^{11} - 10 p^{4} T^{12} - 14 p^{5} T^{13} - p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 7 | \( ( 1 + 4 T + 27 T^{2} + 69 T^{3} + 272 T^{4} + 69 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 + 7 T - 10 T^{2} - 87 T^{3} + 551 T^{4} + 1480 T^{5} - 9928 T^{6} - 3324 T^{7} + 178356 T^{8} - 3324 p T^{9} - 9928 p^{2} T^{10} + 1480 p^{3} T^{11} + 551 p^{4} T^{12} - 87 p^{5} T^{13} - 10 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T - 37 T^{2} + 98 T^{3} + 554 T^{4} - 2096 T^{5} - 5804 T^{6} + 945 p T^{7} + 106065 T^{8} + 945 p^{2} T^{9} - 5804 p^{2} T^{10} - 2096 p^{3} T^{11} + 554 p^{4} T^{12} + 98 p^{5} T^{13} - 37 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T - 71 T^{2} - 140 T^{3} + 2937 T^{4} + 4618 T^{5} - 86145 T^{6} - 48856 T^{7} + 2108699 T^{8} - 48856 p T^{9} - 86145 p^{2} T^{10} + 4618 p^{3} T^{11} + 2937 p^{4} T^{12} - 140 p^{5} T^{13} - 71 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - T - 52 T^{2} + 509 T^{3} + 1277 T^{4} - 20540 T^{5} + 93655 T^{6} + 443301 T^{7} - 3745740 T^{8} + 443301 p T^{9} + 93655 p^{2} T^{10} - 20540 p^{3} T^{11} + 1277 p^{4} T^{12} + 509 p^{5} T^{13} - 52 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 57 T^{2} - 5 T^{3} + 2675 T^{4} - 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 2 T + 117 T^{2} + 99 T^{3} + 5802 T^{4} + 99 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 8 T - 13 T^{2} - 590 T^{3} + 5111 T^{4} + 9890 T^{5} + 199801 T^{6} - 1514064 T^{7} - 3467655 T^{8} - 1514064 p T^{9} + 199801 p^{2} T^{10} + 9890 p^{3} T^{11} + 5111 p^{4} T^{12} - 590 p^{5} T^{13} - 13 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + T - 73 T^{2} + 210 T^{3} + 2084 T^{4} - 14876 T^{5} + 25160 T^{6} + 413805 T^{7} - 1612143 T^{8} + 413805 p T^{9} + 25160 p^{2} T^{10} - 14876 p^{3} T^{11} + 2084 p^{4} T^{12} + 210 p^{5} T^{13} - 73 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 12 T + 11 T^{2} - 70 T^{3} + 2765 T^{4} + 1092 T^{5} - 27713 T^{6} + 366602 T^{7} - 8145777 T^{8} + 366602 p T^{9} - 27713 p^{2} T^{10} + 1092 p^{3} T^{11} + 2765 p^{4} T^{12} - 70 p^{5} T^{13} + 11 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 5 T - 101 T^{2} + 506 T^{3} + 5352 T^{4} - 24832 T^{5} - 88212 T^{6} + 707641 T^{7} - 3097531 T^{8} + 707641 p T^{9} - 88212 p^{2} T^{10} - 24832 p^{3} T^{11} + 5352 p^{4} T^{12} + 506 p^{5} T^{13} - 101 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 5 T - 86 T^{2} + 215 T^{3} + 2385 T^{4} + 14660 T^{5} - 81789 T^{6} - 1000355 T^{7} + 10693034 T^{8} - 1000355 p T^{9} - 81789 p^{2} T^{10} + 14660 p^{3} T^{11} + 2385 p^{4} T^{12} + 215 p^{5} T^{13} - 86 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 114 T^{2} + 176 T^{3} + 3481 T^{4} - 15400 T^{5} - 228578 T^{6} + 386584 T^{7} + 27858884 T^{8} + 386584 p T^{9} - 228578 p^{2} T^{10} - 15400 p^{3} T^{11} + 3481 p^{4} T^{12} + 176 p^{5} T^{13} - 114 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 + 4 T - 216 T^{2} - 728 T^{3} + 28338 T^{4} + 70180 T^{5} - 2597792 T^{6} - 2011948 T^{7} + 193609507 T^{8} - 2011948 p T^{9} - 2597792 p^{2} T^{10} + 70180 p^{3} T^{11} + 28338 p^{4} T^{12} - 728 p^{5} T^{13} - 216 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 20 T + 25 T^{2} - 830 T^{3} + 14400 T^{4} + 140590 T^{5} - 883200 T^{6} + 1962095 T^{7} + 162929144 T^{8} + 1962095 p T^{9} - 883200 p^{2} T^{10} + 140590 p^{3} T^{11} + 14400 p^{4} T^{12} - 830 p^{5} T^{13} + 25 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 20 T + 13 T^{2} + 582 T^{3} + 22431 T^{4} - 149340 T^{5} - 1710159 T^{6} + 34288 p T^{7} + 167178049 T^{8} + 34288 p^{2} T^{9} - 1710159 p^{2} T^{10} - 149340 p^{3} T^{11} + 22431 p^{4} T^{12} + 582 p^{5} T^{13} + 13 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 17 T - 99 T^{2} - 18 p T^{3} + 33753 T^{4} + 218511 T^{5} - 3340534 T^{6} - 1230695 T^{7} + 402442002 T^{8} - 1230695 p T^{9} - 3340534 p^{2} T^{10} + 218511 p^{3} T^{11} + 33753 p^{4} T^{12} - 18 p^{6} T^{13} - 99 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - T + 270 T^{2} - 194 T^{3} + 31408 T^{4} - 194 p T^{5} + 270 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 11 T - 145 T^{2} - 172 T^{3} + 26027 T^{4} - 59285 T^{5} - 2268638 T^{6} + 481263 T^{7} + 100591392 T^{8} + 481263 p T^{9} - 2268638 p^{2} T^{10} - 59285 p^{3} T^{11} + 26027 p^{4} T^{12} - 172 p^{5} T^{13} - 145 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + T - 121 T^{2} - 2070 T^{3} + 1496 T^{4} + 215440 T^{5} + 1891076 T^{6} - 10407525 T^{7} - 192172179 T^{8} - 10407525 p T^{9} + 1891076 p^{2} T^{10} + 215440 p^{3} T^{11} + 1496 p^{4} T^{12} - 2070 p^{5} T^{13} - 121 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.72366172498670331588153497256, −6.72217172471589758671393641226, −6.51440827645734283388967561323, −6.05154862647767140795018180846, −6.01627479357606074512520245157, −5.82261536358874283246359101200, −5.58188531409221067478323579495, −5.45370201816457705828551160740, −5.24124426899623972660220834703, −4.88222841823786374027609550257, −4.83039531908347741928552941327, −4.73530232155458301232763742918, −4.40739101166759622335440238493, −4.26193114455139473223999618631, −4.21826795674787904797372144880, −3.78827791908113804786578006591, −3.46472210352320493045315886827, −3.43145695284455644466045054413, −3.30775523632164727215163005491, −2.74659418026308953269554871505, −2.60848274395953931291229738353, −2.34737899186682024585389260596, −1.63166237457043358887318819985, −1.52314459379814844292666739878, −0.55146671449796190213619435113, 0.55146671449796190213619435113, 1.52314459379814844292666739878, 1.63166237457043358887318819985, 2.34737899186682024585389260596, 2.60848274395953931291229738353, 2.74659418026308953269554871505, 3.30775523632164727215163005491, 3.43145695284455644466045054413, 3.46472210352320493045315886827, 3.78827791908113804786578006591, 4.21826795674787904797372144880, 4.26193114455139473223999618631, 4.40739101166759622335440238493, 4.73530232155458301232763742918, 4.83039531908347741928552941327, 4.88222841823786374027609550257, 5.24124426899623972660220834703, 5.45370201816457705828551160740, 5.58188531409221067478323579495, 5.82261536358874283246359101200, 6.01627479357606074512520245157, 6.05154862647767140795018180846, 6.51440827645734283388967561323, 6.72217172471589758671393641226, 6.72366172498670331588153497256
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.