Dirichlet series
| L(s) = 1 | + 2-s + 3-s + 2·4-s − 2·5-s + 6-s − 8·7-s + 2·8-s + 6·9-s − 2·10-s + 2·12-s − 11·13-s − 8·14-s − 2·15-s + 9·16-s − 4·17-s + 6·18-s − 11·19-s − 4·20-s − 8·21-s − 18·23-s + 2·24-s + 25-s − 11·26-s + 3·27-s − 16·28-s − 11·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.408·6-s − 3.02·7-s + 0.707·8-s + 2·9-s − 0.632·10-s + 0.577·12-s − 3.05·13-s − 2.13·14-s − 0.516·15-s + 9/4·16-s − 0.970·17-s + 1.41·18-s − 2.52·19-s − 0.894·20-s − 1.74·21-s − 3.75·23-s + 0.408·24-s + 1/5·25-s − 2.15·26-s + 0.577·27-s − 3.02·28-s − 2.04·29-s − 0.365·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(296660.\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.038577881\) |
| \(L(\frac12)\) | \(\approx\) | \(1.038577881\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T - T^{2} + T^{3} - 3 p T^{4} + 11 T^{5} + 3 T^{6} - 17 T^{7} + 23 T^{8} - 17 p T^{9} + 3 p^{2} T^{10} + 11 p^{3} T^{11} - 3 p^{5} T^{12} + p^{5} T^{13} - p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - T - 5 T^{2} + 8 T^{3} + T^{4} - 44 T^{5} + 74 T^{6} + 25 p T^{7} - 311 T^{8} + 25 p^{2} T^{9} + 74 p^{2} T^{10} - 44 p^{3} T^{11} + p^{4} T^{12} + 8 p^{5} T^{13} - 5 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 8 T + 25 T^{2} + 29 T^{3} - 59 T^{4} - 263 T^{5} + 31 T^{6} + 2350 T^{7} + 8159 T^{8} + 2350 p T^{9} + 31 p^{2} T^{10} - 263 p^{3} T^{11} - 59 p^{4} T^{12} + 29 p^{5} T^{13} + 25 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 11 T + 27 T^{2} - 129 T^{3} - 773 T^{4} - 646 T^{5} + 3936 T^{6} + 10652 T^{7} + 16527 T^{8} + 10652 p T^{9} + 3936 p^{2} T^{10} - 646 p^{3} T^{11} - 773 p^{4} T^{12} - 129 p^{5} T^{13} + 27 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T - 11 T^{2} - 44 T^{3} + 474 T^{4} + 576 T^{5} - 13347 T^{6} - 12022 T^{7} + 165123 T^{8} - 12022 p T^{9} - 13347 p^{2} T^{10} + 576 p^{3} T^{11} + 474 p^{4} T^{12} - 44 p^{5} T^{13} - 11 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 11 T + 18 T^{2} - 296 T^{3} - 98 p T^{4} - 3809 T^{5} + 5960 T^{6} + 104090 T^{7} + 621431 T^{8} + 104090 p T^{9} + 5960 p^{2} T^{10} - 3809 p^{3} T^{11} - 98 p^{5} T^{12} - 296 p^{5} T^{13} + 18 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 9 T + 38 T^{2} - 85 T^{3} - 979 T^{4} - 85 p T^{5} + 38 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 11 T + 48 T^{2} + 434 T^{3} + 4878 T^{4} + 28451 T^{5} + 129000 T^{6} + 940640 T^{7} + 6410691 T^{8} + 940640 p T^{9} + 129000 p^{2} T^{10} + 28451 p^{3} T^{11} + 4878 p^{4} T^{12} + 434 p^{5} T^{13} + 48 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 9 T - 25 T^{2} - 375 T^{3} + 125 T^{4} + 11172 T^{5} + 53248 T^{6} - 180600 T^{7} - 3080375 T^{8} - 180600 p T^{9} + 53248 p^{2} T^{10} + 11172 p^{3} T^{11} + 125 p^{4} T^{12} - 375 p^{5} T^{13} - 25 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T + 18 T^{2} - 73 T^{3} + 2596 T^{4} - 6424 T^{5} - 4270 T^{6} - 387044 T^{7} + 3296339 T^{8} - 387044 p T^{9} - 4270 p^{2} T^{10} - 6424 p^{3} T^{11} + 2596 p^{4} T^{12} - 73 p^{5} T^{13} + 18 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 11 T + 60 T^{2} - 365 T^{3} + 4110 T^{4} - 18518 T^{5} + 48 p T^{6} - 27020 T^{7} + 2958855 T^{8} - 27020 p T^{9} + 48 p^{3} T^{10} - 18518 p^{3} T^{11} + 4110 p^{4} T^{12} - 365 p^{5} T^{13} + 60 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 21 T + 293 T^{2} - 2900 T^{3} + 21441 T^{4} - 2900 p T^{5} + 293 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + T - 117 T^{2} + 67 T^{3} + 3881 T^{4} - 10684 T^{5} + 239850 T^{6} + 373586 T^{7} - 23291731 T^{8} + 373586 p T^{9} + 239850 p^{2} T^{10} - 10684 p^{3} T^{11} + 3881 p^{4} T^{12} + 67 p^{5} T^{13} - 117 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T - 79 T^{2} + 19 p T^{3} + 1289 T^{4} - 24517 T^{5} - 106353 T^{6} - 128744 T^{7} + 15887493 T^{8} - 128744 p T^{9} - 106353 p^{2} T^{10} - 24517 p^{3} T^{11} + 1289 p^{4} T^{12} + 19 p^{6} T^{13} - 79 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 26 T + 333 T^{2} - 2939 T^{3} + 20403 T^{4} - 42071 T^{5} - 1170525 T^{6} + 17574820 T^{7} - 151981089 T^{8} + 17574820 p T^{9} - 1170525 p^{2} T^{10} - 42071 p^{3} T^{11} + 20403 p^{4} T^{12} - 2939 p^{5} T^{13} + 333 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 2 T - 143 T^{2} - 458 T^{3} + 6653 T^{4} + 35302 T^{5} + 301955 T^{6} - 1069630 T^{7} - 43463684 T^{8} - 1069630 p T^{9} + 301955 p^{2} T^{10} + 35302 p^{3} T^{11} + 6653 p^{4} T^{12} - 458 p^{5} T^{13} - 143 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 25 T + 312 T^{2} + 3635 T^{3} + 42358 T^{4} + 443170 T^{5} + 4435074 T^{6} + 38435370 T^{7} + 307329585 T^{8} + 38435370 p T^{9} + 4435074 p^{2} T^{10} + 443170 p^{3} T^{11} + 42358 p^{4} T^{12} + 3635 p^{5} T^{13} + 312 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 32 T + 377 T^{2} - 2166 T^{3} + 12476 T^{4} - 81202 T^{5} - 135735 T^{6} + 2520912 T^{7} + 5093859 T^{8} + 2520912 p T^{9} - 135735 p^{2} T^{10} - 81202 p^{3} T^{11} + 12476 p^{4} T^{12} - 2166 p^{5} T^{13} + 377 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 23 T + 211 T^{2} + 1356 T^{3} + 14925 T^{4} + 22928 T^{5} - 1975610 T^{6} - 22147285 T^{7} - 149759579 T^{8} - 22147285 p T^{9} - 1975610 p^{2} T^{10} + 22928 p^{3} T^{11} + 14925 p^{4} T^{12} + 1356 p^{5} T^{13} + 211 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 10 T - 5 T^{2} - 955 T^{3} + 24141 T^{4} - 156705 T^{5} + 82125 T^{6} - 12021140 T^{7} + 246252711 T^{8} - 12021140 p T^{9} + 82125 p^{2} T^{10} - 156705 p^{3} T^{11} + 24141 p^{4} T^{12} - 955 p^{5} T^{13} - 5 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 206 T^{2} - 400 T^{3} + 21551 T^{4} - 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 18 T - 65 T^{2} - 2286 T^{3} + 4086 T^{4} + 224892 T^{5} + 1296371 T^{6} - 2682270 T^{7} - 108522781 T^{8} - 2682270 p T^{9} + 1296371 p^{2} T^{10} + 224892 p^{3} T^{11} + 4086 p^{4} T^{12} - 2286 p^{5} T^{13} - 65 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.62121786445294360211852377250, −4.34480881724596366347082329615, −4.19386031593854187078001167991, −4.17616013334000436309583217884, −3.93829693321125572072572074472, −3.93265379115151512710304969610, −3.88014314190586112035824252350, −3.79430728235839605845852273783, −3.68086051413668763614459375436, −3.56615378708815446646978775221, −3.24109342093974909557958607225, −3.03274890250287252281945366698, −2.78594216815440608234030656357, −2.76322330985091459721988734084, −2.47997483266997600599060637031, −2.39822911128922176567782582556, −2.18120967046739692917107311782, −2.16515163151313818508549498812, −2.09905235721235898811180792404, −1.92764647494473629218231544609, −1.63964189851892082440002864301, −1.04561559098535475080584348046, −0.961068608546214148341095418066, −0.35554581291674849780298409028, −0.24444803778015019238059081238, 0.24444803778015019238059081238, 0.35554581291674849780298409028, 0.961068608546214148341095418066, 1.04561559098535475080584348046, 1.63964189851892082440002864301, 1.92764647494473629218231544609, 2.09905235721235898811180792404, 2.16515163151313818508549498812, 2.18120967046739692917107311782, 2.39822911128922176567782582556, 2.47997483266997600599060637031, 2.76322330985091459721988734084, 2.78594216815440608234030656357, 3.03274890250287252281945366698, 3.24109342093974909557958607225, 3.56615378708815446646978775221, 3.68086051413668763614459375436, 3.79430728235839605845852273783, 3.88014314190586112035824252350, 3.93265379115151512710304969610, 3.93829693321125572072572074472, 4.17616013334000436309583217884, 4.19386031593854187078001167991, 4.34480881724596366347082329615, 4.62121786445294360211852377250
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.