Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 8·5-s − 4·6-s − 2·7-s + 4·8-s + 5·9-s + 16·10-s − 2·11-s − 6·12-s − 2·13-s − 4·14-s − 16·15-s + 4·16-s − 6·17-s + 10·18-s − 6·19-s + 24·20-s + 4·21-s − 4·22-s − 8·23-s − 8·24-s − 4·25-s − 4·26-s − 6·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 3.57·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 5/3·9-s + 5.05·10-s − 0.603·11-s − 1.73·12-s − 0.554·13-s − 1.06·14-s − 4.13·15-s + 16-s − 1.45·17-s + 2.35·18-s − 1.37·19-s + 5.36·20-s + 0.872·21-s − 0.852·22-s − 1.66·23-s − 1.63·24-s − 4/5·25-s − 0.784·26-s − 1.15·27-s − 1.13·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.20227\times 10^{7}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 31^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9163777444\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9163777444\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - p T + T^{2} + T^{4} - p^{4} T^{5} + 25 T^{6} - 3 p T^{7} - 3 T^{8} - 3 p^{2} T^{9} + 25 p^{2} T^{10} - p^{7} T^{11} + p^{4} T^{12} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + 2 T - T^{2} - 2 p T^{3} - 4 T^{4} + 4 p^{2} T^{5} + 65 T^{6} - 20 T^{7} - 113 T^{8} - 20 p T^{9} + 65 p^{2} T^{10} + 4 p^{5} T^{11} - 4 p^{4} T^{12} - 2 p^{6} T^{13} - p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 5 | \( ( 1 - T + p T^{2} )^{8} \) | |
| 7 | \( 1 + 2 T - 9 T^{2} - 30 T^{3} + 36 T^{4} + 156 T^{5} - 135 T^{6} - 324 T^{7} + 1527 T^{8} - 324 p T^{9} - 135 p^{2} T^{10} + 156 p^{3} T^{11} + 36 p^{4} T^{12} - 30 p^{5} T^{13} - 9 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T - T^{2} + 10 T^{3} - 52 T^{4} + 820 T^{5} + 2241 T^{6} - 1972 T^{7} + 9183 T^{8} - 1972 p T^{9} + 2241 p^{2} T^{10} + 820 p^{3} T^{11} - 52 p^{4} T^{12} + 10 p^{5} T^{13} - p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T - 15 T^{2} - 42 T^{3} + 84 T^{4} + 996 T^{5} + 1839 T^{6} - 5964 T^{7} - 35169 T^{8} - 5964 p T^{9} + 1839 p^{2} T^{10} + 996 p^{3} T^{11} + 84 p^{4} T^{12} - 42 p^{5} T^{13} - 15 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T + T^{2} - 6 p T^{3} - 324 T^{4} + 84 p T^{5} + 9215 T^{6} + 1740 T^{7} - 72793 T^{8} + 1740 p T^{9} + 9215 p^{2} T^{10} + 84 p^{4} T^{11} - 324 p^{4} T^{12} - 6 p^{6} T^{13} + p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 6 T - 9 T^{2} - 210 T^{3} - 28 p T^{4} - 900 T^{5} - 2151 T^{6} + 42756 T^{7} + 442783 T^{8} + 42756 p T^{9} - 2151 p^{2} T^{10} - 900 p^{3} T^{11} - 28 p^{5} T^{12} - 210 p^{5} T^{13} - 9 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 4 T - 7 T^{2} - 120 T^{3} - 319 T^{4} - 120 p T^{5} - 7 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 8 T - 2 T^{2} - 312 T^{3} - 1349 T^{4} + 520 T^{5} + 22492 T^{6} + 95040 T^{7} + 530997 T^{8} + 95040 p T^{9} + 22492 p^{2} T^{10} + 520 p^{3} T^{11} - 1349 p^{4} T^{12} - 312 p^{5} T^{13} - 2 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 + T + p T^{2} )^{8} \) | |
| 41 | \( 1 + 2 T - 7 T^{2} + 46 T^{3} - 1348 T^{4} + 11740 T^{5} + 53847 T^{6} - 209308 T^{7} + 2217735 T^{8} - 209308 p T^{9} + 53847 p^{2} T^{10} + 11740 p^{3} T^{11} - 1348 p^{4} T^{12} + 46 p^{5} T^{13} - 7 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 2 T + 15 T^{2} + 138 T^{3} - 1236 T^{4} + 6996 T^{5} - 15471 T^{6} - 315444 T^{7} + 2085951 T^{8} - 315444 p T^{9} - 15471 p^{2} T^{10} + 6996 p^{3} T^{11} - 1236 p^{4} T^{12} + 138 p^{5} T^{13} + 15 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 8 T - 14 T^{2} - 360 T^{3} - 989 T^{4} + 35464 T^{5} + 256420 T^{6} - 291456 T^{7} - 6813243 T^{8} - 291456 p T^{9} + 256420 p^{2} T^{10} + 35464 p^{3} T^{11} - 989 p^{4} T^{12} - 360 p^{5} T^{13} - 14 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T - 71 T^{2} + 750 T^{3} + 2196 T^{4} - 15468 T^{5} - 181225 T^{6} - 62652 T^{7} + 18517247 T^{8} - 62652 p T^{9} - 181225 p^{2} T^{10} - 15468 p^{3} T^{11} + 2196 p^{4} T^{12} + 750 p^{5} T^{13} - 71 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 6 T - 41 T^{2} + 6 p T^{3} - 324 T^{4} - 1092 p T^{5} + 428921 T^{6} + 1269852 T^{7} - 16710673 T^{8} + 1269852 p T^{9} + 428921 p^{2} T^{10} - 1092 p^{4} T^{11} - 324 p^{4} T^{12} + 6 p^{6} T^{13} - 41 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{4} \) | |
| 67 | \( ( 1 + 2 T + 117 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 71 | \( 1 - 14 T + 55 T^{2} + 210 T^{3} - 1820 T^{4} - 72772 T^{5} + 789433 T^{6} - 40740 T^{7} - 29229705 T^{8} - 40740 p T^{9} + 789433 p^{2} T^{10} - 72772 p^{3} T^{11} - 1820 p^{4} T^{12} + 210 p^{5} T^{13} + 55 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 2 T - 135 T^{2} + 402 T^{3} + 12924 T^{4} - 15516 T^{5} - 1104681 T^{6} + 336924 T^{7} + 86270151 T^{8} + 336924 p T^{9} - 1104681 p^{2} T^{10} - 15516 p^{3} T^{11} + 12924 p^{4} T^{12} + 402 p^{5} T^{13} - 135 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 22 T + 223 T^{2} + 902 T^{3} - 6364 T^{4} - 127820 T^{5} - 975103 T^{6} - 4781260 T^{7} - 33119273 T^{8} - 4781260 p T^{9} - 975103 p^{2} T^{10} - 127820 p^{3} T^{11} - 6364 p^{4} T^{12} + 902 p^{5} T^{13} + 223 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 6 T - 89 T^{2} - 786 T^{3} + 2508 T^{4} + 114396 T^{5} + 594569 T^{6} - 4068348 T^{7} - 59039617 T^{8} - 4068348 p T^{9} + 594569 p^{2} T^{10} + 114396 p^{3} T^{11} + 2508 p^{4} T^{12} - 786 p^{5} T^{13} - 89 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 8 T - 58 T^{2} - 728 T^{3} - 973 T^{4} + 183880 T^{5} + 1530828 T^{6} - 5113024 T^{7} - 89035515 T^{8} - 5113024 p T^{9} + 1530828 p^{2} T^{10} + 183880 p^{3} T^{11} - 973 p^{4} T^{12} - 728 p^{5} T^{13} - 58 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 16 T + 6 T^{2} - 2352 T^{3} - 23709 T^{4} - 154992 T^{5} - 259380 T^{6} + 19931520 T^{7} + 366279717 T^{8} + 19931520 p T^{9} - 259380 p^{2} T^{10} - 154992 p^{3} T^{11} - 23709 p^{4} T^{12} - 2352 p^{5} T^{13} + 6 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.35796226171341044327663549902, −4.14788155802545182707558147771, −4.02364988197992490867794547603, −3.96343562414858544794805002969, −3.86417833872985660477945235632, −3.83929580130220073589804698168, −3.78738520797133276492607401181, −3.32752993579582653685878773543, −3.26912413271597384975892766400, −2.93664624550242929592999052010, −2.91045123429347156003971823042, −2.77818011815767627582475851594, −2.38779384657354344961201066707, −2.33920794356340236606440686716, −2.27217038297573760227989382913, −2.27064705210076804052635103640, −1.97528934845772330902290957294, −1.95791126517290227916977239341, −1.76973640119156840354323532387, −1.58562171812125841949779020464, −1.57252130922744678677997649984, −1.26904479384264766825004381611, −0.926999746360359762691088287768, −0.28218391925381551714308506094, −0.12872691484090804017722086150, 0.12872691484090804017722086150, 0.28218391925381551714308506094, 0.926999746360359762691088287768, 1.26904479384264766825004381611, 1.57252130922744678677997649984, 1.58562171812125841949779020464, 1.76973640119156840354323532387, 1.95791126517290227916977239341, 1.97528934845772330902290957294, 2.27064705210076804052635103640, 2.27217038297573760227989382913, 2.33920794356340236606440686716, 2.38779384657354344961201066707, 2.77818011815767627582475851594, 2.91045123429347156003971823042, 2.93664624550242929592999052010, 3.26912413271597384975892766400, 3.32752993579582653685878773543, 3.78738520797133276492607401181, 3.83929580130220073589804698168, 3.86417833872985660477945235632, 3.96343562414858544794805002969, 4.02364988197992490867794547603, 4.14788155802545182707558147771, 4.35796226171341044327663549902
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.