Dirichlet series
| L(s) = 1 | − 8·2-s + 32·4-s + 6·5-s − 2·7-s − 80·8-s − 48·10-s + 8·11-s + 6·13-s + 16·14-s + 120·16-s − 4·17-s + 192·20-s − 64·22-s − 4·23-s + 11·25-s − 48·26-s − 64·28-s + 32·31-s − 32·32-s + 32·34-s − 12·35-s + 60·37-s − 480·40-s − 184·41-s + 126·43-s + 256·44-s + 32·46-s + ⋯ |
| L(s) = 1 | − 4·2-s + 8·4-s + 6/5·5-s − 2/7·7-s − 10·8-s − 4.79·10-s + 8/11·11-s + 6/13·13-s + 8/7·14-s + 15/2·16-s − 0.235·17-s + 48/5·20-s − 2.90·22-s − 0.173·23-s + 0.439·25-s − 1.84·26-s − 2.28·28-s + 1.03·31-s − 32-s + 0.941·34-s − 0.342·35-s + 1.62·37-s − 12·40-s − 4.48·41-s + 2.93·43-s + 5.81·44-s + 0.695·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.63067\times 10^{10}\) |
| Root analytic conductor: | \(4.69796\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.001840310669\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001840310669\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 6 T + p^{2} T^{2} - 48 T^{3} - 96 p T^{4} - 48 p^{2} T^{5} + p^{6} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 7 | \( 1 + 2 T + 2 T^{2} - 302 T^{3} - 3964 T^{4} - 17930 T^{5} + 17670 T^{6} + 772230 T^{7} + 15412870 T^{8} + 772230 p^{2} T^{9} + 17670 p^{4} T^{10} - 17930 p^{6} T^{11} - 3964 p^{8} T^{12} - 302 p^{10} T^{13} + 2 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 4 T + 139 T^{2} - 952 T^{3} + 29356 T^{4} - 952 p^{2} T^{5} + 139 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 13 | \( 1 - 6 T + 18 T^{2} + 2220 T^{3} + 8377 T^{4} - 130860 T^{5} + 3098574 T^{6} + 70477302 T^{7} - 1160962800 T^{8} + 70477302 p^{2} T^{9} + 3098574 p^{4} T^{10} - 130860 p^{6} T^{11} + 8377 p^{8} T^{12} + 2220 p^{10} T^{13} + 18 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 4 T + 8 T^{2} + 10622 T^{3} + 63065 T^{4} - 2358760 T^{5} + 46473882 T^{6} + 328441602 T^{7} - 20270113712 T^{8} + 328441602 p^{2} T^{9} + 46473882 p^{4} T^{10} - 2358760 p^{6} T^{11} + 63065 p^{8} T^{12} + 10622 p^{10} T^{13} + 8 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 1294 T^{2} + 801913 T^{4} - 341123926 T^{6} + 126209320372 T^{8} - 341123926 p^{4} T^{10} + 801913 p^{8} T^{12} - 1294 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 + 4 T + 8 T^{2} - 6484 T^{3} + 65264 T^{4} + 1219892 T^{5} + 25378584 T^{6} - 1644669732 T^{7} - 103786324514 T^{8} - 1644669732 p^{2} T^{9} + 25378584 p^{4} T^{10} + 1219892 p^{6} T^{11} + 65264 p^{8} T^{12} - 6484 p^{10} T^{13} + 8 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 4120 T^{2} + 8288266 T^{4} - 10848831280 T^{6} + 10438249323211 T^{8} - 10848831280 p^{4} T^{10} + 8288266 p^{8} T^{12} - 4120 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 - 16 T + 931 T^{2} - 5008 T^{3} + 1404148 T^{4} - 5008 p^{2} T^{5} + 931 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 - 60 T + 1800 T^{2} - 41478 T^{3} - 1861271 T^{4} + 98480448 T^{5} - 1698326838 T^{6} - 4874606946 T^{7} + 3040933285296 T^{8} - 4874606946 p^{2} T^{9} - 1698326838 p^{4} T^{10} + 98480448 p^{6} T^{11} - 1861271 p^{8} T^{12} - 41478 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 + 92 T + 6031 T^{2} + 220148 T^{3} + 9418876 T^{4} + 220148 p^{2} T^{5} + 6031 p^{4} T^{6} + 92 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 - 126 T + 7938 T^{2} - 381846 T^{3} + 10147876 T^{4} + 147906846 T^{5} - 26286918426 T^{6} + 1764453046182 T^{7} - 90357353577210 T^{8} + 1764453046182 p^{2} T^{9} - 26286918426 p^{4} T^{10} + 147906846 p^{6} T^{11} + 10147876 p^{8} T^{12} - 381846 p^{10} T^{13} + 7938 p^{12} T^{14} - 126 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 150 T + 11250 T^{2} + 799350 T^{3} + 59734276 T^{4} + 3545114850 T^{5} + 179236833750 T^{6} + 9629153539650 T^{7} + 496924665634566 T^{8} + 9629153539650 p^{2} T^{9} + 179236833750 p^{4} T^{10} + 3545114850 p^{6} T^{11} + 59734276 p^{8} T^{12} + 799350 p^{10} T^{13} + 11250 p^{12} T^{14} + 150 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 238 T + 28322 T^{2} - 44902 p T^{3} + 139930688 T^{4} - 4566717266 T^{5} - 44499937410 T^{6} + 17953679213214 T^{7} - 1331953112777282 T^{8} + 17953679213214 p^{2} T^{9} - 44499937410 p^{4} T^{10} - 4566717266 p^{6} T^{11} + 139930688 p^{8} T^{12} - 44902 p^{11} T^{13} + 28322 p^{12} T^{14} - 238 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 14230 T^{2} + 116898241 T^{4} - 643494544270 T^{6} + 2613655452497956 T^{8} - 643494544270 p^{4} T^{10} + 116898241 p^{8} T^{12} - 14230 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 + 84 T + 13129 T^{2} + 781824 T^{3} + 68365056 T^{4} + 781824 p^{2} T^{5} + 13129 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 + 128 T + 8192 T^{2} + 656272 T^{3} + 50551712 T^{4} + 2693240560 T^{5} + 145961635968 T^{6} + 9558058471296 T^{7} + 619084627316350 T^{8} + 9558058471296 p^{2} T^{9} + 145961635968 p^{4} T^{10} + 2693240560 p^{6} T^{11} + 50551712 p^{8} T^{12} + 656272 p^{10} T^{13} + 8192 p^{12} T^{14} + 128 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 214 T + 25873 T^{2} + 2624806 T^{3} + 217648792 T^{4} + 2624806 p^{2} T^{5} + 25873 p^{4} T^{6} + 214 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 44 T + 968 T^{2} - 168838 T^{3} - 14816503 T^{4} + 1116791144 T^{5} - 20543300310 T^{6} + 3054684235086 T^{7} - 340260294902192 T^{8} + 3054684235086 p^{2} T^{9} - 20543300310 p^{4} T^{10} + 1116791144 p^{6} T^{11} - 14816503 p^{8} T^{12} - 168838 p^{10} T^{13} + 968 p^{12} T^{14} - 44 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 36068 T^{2} + 618081796 T^{4} - 6652199280668 T^{6} + 49385218968591094 T^{8} - 6652199280668 p^{4} T^{10} + 618081796 p^{8} T^{12} - 36068 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 + 150 T + 11250 T^{2} + 1154982 T^{3} + 132173380 T^{4} + 10744540290 T^{5} + 791722228662 T^{6} + 87037745962866 T^{7} + 9419701961427654 T^{8} + 87037745962866 p^{2} T^{9} + 791722228662 p^{4} T^{10} + 10744540290 p^{6} T^{11} + 132173380 p^{8} T^{12} + 1154982 p^{10} T^{13} + 11250 p^{12} T^{14} + 150 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 33568 T^{2} + 508725346 T^{4} - 5114710147648 T^{6} + 42682260072542179 T^{8} - 5114710147648 p^{4} T^{10} + 508725346 p^{8} T^{12} - 33568 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 - 210 T + 22050 T^{2} - 2333154 T^{3} + 229616224 T^{4} - 19978947774 T^{5} + 1854345087198 T^{6} - 188100400226958 T^{7} + 19054899041397054 T^{8} - 188100400226958 p^{2} T^{9} + 1854345087198 p^{4} T^{10} - 19978947774 p^{6} T^{11} + 229616224 p^{8} T^{12} - 2333154 p^{10} T^{13} + 22050 p^{12} T^{14} - 210 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−4.18840805423589571174957024839, −4.15685507096514319146222624410, −3.84579798033021372480462671155, −3.76737857398908774255587284524, −3.53039893833532280559455762407, −3.47467183602666027641576610968, −3.22368046360054084499358367127, −2.99755026919780580733671557524, −2.94904486330276193677262243231, −2.89303639010837141236968804166, −2.64670529886142491694559088737, −2.42496118557869481354047209941, −2.32800802266835023348894179166, −2.27891753471660117540703316707, −1.85277394642224875623086678913, −1.77920816152297165482355579543, −1.68976531498862687417792118487, −1.55660085663145507596198088435, −1.19324214086766509940995203567, −1.19047395420177967693098761246, −1.04347857085866492583886481578, −1.03507882533393516085689131202, −0.45823110155291064609430256644, −0.25196944344874404263483993972, −0.01489107187604259010679138821, 0.01489107187604259010679138821, 0.25196944344874404263483993972, 0.45823110155291064609430256644, 1.03507882533393516085689131202, 1.04347857085866492583886481578, 1.19047395420177967693098761246, 1.19324214086766509940995203567, 1.55660085663145507596198088435, 1.68976531498862687417792118487, 1.77920816152297165482355579543, 1.85277394642224875623086678913, 2.27891753471660117540703316707, 2.32800802266835023348894179166, 2.42496118557869481354047209941, 2.64670529886142491694559088737, 2.89303639010837141236968804166, 2.94904486330276193677262243231, 2.99755026919780580733671557524, 3.22368046360054084499358367127, 3.47467183602666027641576610968, 3.53039893833532280559455762407, 3.76737857398908774255587284524, 3.84579798033021372480462671155, 4.15685507096514319146222624410, 4.18840805423589571174957024839
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.