Dirichlet series
| L(s) = 1 | + 8·2-s + 8·3-s + 36·4-s − 8·5-s + 64·6-s − 6·7-s + 120·8-s + 36·9-s − 64·10-s − 7·11-s + 288·12-s + 8·13-s − 48·14-s − 64·15-s + 330·16-s − 20·17-s + 288·18-s − 12·19-s − 288·20-s − 48·21-s − 56·22-s − 14·23-s + 960·24-s + 11·25-s + 64·26-s + 120·27-s − 216·28-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 4.61·3-s + 18·4-s − 3.57·5-s + 26.1·6-s − 2.26·7-s + 42.4·8-s + 12·9-s − 20.2·10-s − 2.11·11-s + 83.1·12-s + 2.21·13-s − 12.8·14-s − 16.5·15-s + 82.5·16-s − 4.85·17-s + 67.8·18-s − 2.75·19-s − 64.3·20-s − 10.4·21-s − 11.9·22-s − 2.91·23-s + 195.·24-s + 11/5·25-s + 12.5·26-s + 23.0·27-s − 40.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.86860\times 10^{14}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 - T )^{8} \) | |
| 103 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + 8 T + 53 T^{2} + 233 T^{3} + 901 T^{4} + 2788 T^{5} + 1594 p T^{6} + 19668 T^{7} + 46648 T^{8} + 19668 p T^{9} + 1594 p^{3} T^{10} + 2788 p^{3} T^{11} + 901 p^{4} T^{12} + 233 p^{5} T^{13} + 53 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 6 T + 50 T^{2} + 200 T^{3} + 1003 T^{4} + 3112 T^{5} + 11930 T^{6} + 30677 T^{7} + 98025 T^{8} + 30677 p T^{9} + 11930 p^{2} T^{10} + 3112 p^{3} T^{11} + 1003 p^{4} T^{12} + 200 p^{5} T^{13} + 50 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 7 T + 61 T^{2} + 243 T^{3} + 108 p T^{4} + 3238 T^{5} + 13049 T^{6} + 30939 T^{7} + 134247 T^{8} + 30939 p T^{9} + 13049 p^{2} T^{10} + 3238 p^{3} T^{11} + 108 p^{5} T^{12} + 243 p^{5} T^{13} + 61 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 20 T + 16 p T^{2} + 2619 T^{3} + 20648 T^{4} + 134440 T^{5} + 761559 T^{6} + 3749008 T^{7} + 16461265 T^{8} + 3749008 p T^{9} + 761559 p^{2} T^{10} + 134440 p^{3} T^{11} + 20648 p^{4} T^{12} + 2619 p^{5} T^{13} + 16 p^{7} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 12 T + 144 T^{2} + 1120 T^{3} + 8653 T^{4} + 51782 T^{5} + 304550 T^{6} + 1478705 T^{7} + 7052218 T^{8} + 1478705 p T^{9} + 304550 p^{2} T^{10} + 51782 p^{3} T^{11} + 8653 p^{4} T^{12} + 1120 p^{5} T^{13} + 144 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 14 T + 187 T^{2} + 1683 T^{3} + 13831 T^{4} + 95563 T^{5} + 594575 T^{6} + 3325627 T^{7} + 16676608 T^{8} + 3325627 p T^{9} + 594575 p^{2} T^{10} + 95563 p^{3} T^{11} + 13831 p^{4} T^{12} + 1683 p^{5} T^{13} + 187 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 25 T + 429 T^{2} + 5351 T^{3} + 55139 T^{4} + 473956 T^{5} + 3543400 T^{6} + 23012323 T^{7} + 132287252 T^{8} + 23012323 p T^{9} + 3543400 p^{2} T^{10} + 473956 p^{3} T^{11} + 55139 p^{4} T^{12} + 5351 p^{5} T^{13} + 429 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 12 T + 160 T^{2} + 1020 T^{3} + 8033 T^{4} + 36594 T^{5} + 275710 T^{6} + 1225551 T^{7} + 9308526 T^{8} + 1225551 p T^{9} + 275710 p^{2} T^{10} + 36594 p^{3} T^{11} + 8033 p^{4} T^{12} + 1020 p^{5} T^{13} + 160 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 15 T + 305 T^{2} + 3241 T^{3} + 38479 T^{4} + 316856 T^{5} + 2757678 T^{6} + 18281909 T^{7} + 125872348 T^{8} + 18281909 p T^{9} + 2757678 p^{2} T^{10} + 316856 p^{3} T^{11} + 38479 p^{4} T^{12} + 3241 p^{5} T^{13} + 305 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 18 T + 365 T^{2} + 4210 T^{3} + 49609 T^{4} + 425338 T^{5} + 3714236 T^{6} + 25641027 T^{7} + 182463800 T^{8} + 25641027 p T^{9} + 3714236 p^{2} T^{10} + 425338 p^{3} T^{11} + 49609 p^{4} T^{12} + 4210 p^{5} T^{13} + 365 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T + 252 T^{2} + 1772 T^{3} + 30126 T^{4} + 181972 T^{5} + 2232955 T^{6} + 11553825 T^{7} + 113879774 T^{8} + 11553825 p T^{9} + 2232955 p^{2} T^{10} + 181972 p^{3} T^{11} + 30126 p^{4} T^{12} + 1772 p^{5} T^{13} + 252 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 204 T^{2} + 1420 T^{3} + 16459 T^{4} + 91876 T^{5} + 1020018 T^{6} + 5263987 T^{7} + 53613624 T^{8} + 5263987 p T^{9} + 1020018 p^{2} T^{10} + 91876 p^{3} T^{11} + 16459 p^{4} T^{12} + 1420 p^{5} T^{13} + 204 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 25 T + 540 T^{2} + 7866 T^{3} + 101736 T^{4} + 1073772 T^{5} + 10369999 T^{6} + 86610728 T^{7} + 672940003 T^{8} + 86610728 p T^{9} + 10369999 p^{2} T^{10} + 1073772 p^{3} T^{11} + 101736 p^{4} T^{12} + 7866 p^{5} T^{13} + 540 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 9 T + 297 T^{2} + 2594 T^{3} + 43882 T^{4} + 347012 T^{5} + 4256737 T^{6} + 29163852 T^{7} + 294660638 T^{8} + 29163852 p T^{9} + 4256737 p^{2} T^{10} + 347012 p^{3} T^{11} + 43882 p^{4} T^{12} + 2594 p^{5} T^{13} + 297 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 2 T + 328 T^{2} - 224 T^{3} + 45968 T^{4} - 156822 T^{5} + 3882665 T^{6} - 21401799 T^{7} + 252725776 T^{8} - 21401799 p T^{9} + 3882665 p^{2} T^{10} - 156822 p^{3} T^{11} + 45968 p^{4} T^{12} - 224 p^{5} T^{13} + 328 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 8 T + 413 T^{2} + 2652 T^{3} + 77333 T^{4} + 408459 T^{5} + 8869686 T^{6} + 39292041 T^{7} + 702186539 T^{8} + 39292041 p T^{9} + 8869686 p^{2} T^{10} + 408459 p^{3} T^{11} + 77333 p^{4} T^{12} + 2652 p^{5} T^{13} + 413 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 13 T + 469 T^{2} + 4838 T^{3} + 98597 T^{4} + 833187 T^{5} + 12462947 T^{6} + 87885815 T^{7} + 1060667768 T^{8} + 87885815 p T^{9} + 12462947 p^{2} T^{10} + 833187 p^{3} T^{11} + 98597 p^{4} T^{12} + 4838 p^{5} T^{13} + 469 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 2 T + 158 T^{2} - 884 T^{3} + 11155 T^{4} - 116193 T^{5} + 1328284 T^{6} - 5591553 T^{7} + 124867587 T^{8} - 5591553 p T^{9} + 1328284 p^{2} T^{10} - 116193 p^{3} T^{11} + 11155 p^{4} T^{12} - 884 p^{5} T^{13} + 158 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - T + 255 T^{2} - 1231 T^{3} + 29351 T^{4} - 197950 T^{5} + 3103156 T^{6} - 12293277 T^{7} + 296938044 T^{8} - 12293277 p T^{9} + 3103156 p^{2} T^{10} - 197950 p^{3} T^{11} + 29351 p^{4} T^{12} - 1231 p^{5} T^{13} + 255 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 6 T + 233 T^{2} - 353 T^{3} + 16627 T^{4} - 206133 T^{5} + 1693193 T^{6} - 14679863 T^{7} + 223476998 T^{8} - 14679863 p T^{9} + 1693193 p^{2} T^{10} - 206133 p^{3} T^{11} + 16627 p^{4} T^{12} - 353 p^{5} T^{13} + 233 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 17 T + 577 T^{2} + 8160 T^{3} + 157580 T^{4} + 1840495 T^{5} + 26062700 T^{6} + 251322627 T^{7} + 2830404494 T^{8} + 251322627 p T^{9} + 26062700 p^{2} T^{10} + 1840495 p^{3} T^{11} + 157580 p^{4} T^{12} + 8160 p^{5} T^{13} + 577 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 19 T + 384 T^{2} - 4715 T^{3} + 57951 T^{4} - 602037 T^{5} + 6955159 T^{6} - 76098654 T^{7} + 782771872 T^{8} - 76098654 p T^{9} + 6955159 p^{2} T^{10} - 602037 p^{3} T^{11} + 57951 p^{4} T^{12} - 4715 p^{5} T^{13} + 384 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.75290286048747435048104745653, −3.45513896483081056395977119619, −3.44255684448901276778420646641, −3.40851468180253273826065878659, −3.26921420789251321968306325921, −3.22055317338755301849963008454, −3.14049368476679320259447966174, −3.08039983034068428438875114444, −3.07836190481351074569480004145, −2.54101908987783858501958633561, −2.53153177453051418241098739491, −2.52121476722538677395775615408, −2.47329211667444595612266917206, −2.34445003433395074056130843594, −2.27631051989779732591729489866, −2.27580400915777920317464521794, −2.10377324235724926974740638344, −1.96576134567909217468434000166, −1.63617391401644762056752192713, −1.56638819834020731998080327186, −1.55601438121141562204314350141, −1.55485708642366107505371575788, −1.53533248934244855582213772571, −1.46346008130436752583073201287, −1.42095190742498082818987201034, 0, 0, 0, 0, 0, 0, 0, 0, 1.42095190742498082818987201034, 1.46346008130436752583073201287, 1.53533248934244855582213772571, 1.55485708642366107505371575788, 1.55601438121141562204314350141, 1.56638819834020731998080327186, 1.63617391401644762056752192713, 1.96576134567909217468434000166, 2.10377324235724926974740638344, 2.27580400915777920317464521794, 2.27631051989779732591729489866, 2.34445003433395074056130843594, 2.47329211667444595612266917206, 2.52121476722538677395775615408, 2.53153177453051418241098739491, 2.54101908987783858501958633561, 3.07836190481351074569480004145, 3.08039983034068428438875114444, 3.14049368476679320259447966174, 3.22055317338755301849963008454, 3.26921420789251321968306325921, 3.40851468180253273826065878659, 3.44255684448901276778420646641, 3.45513896483081056395977119619, 3.75290286048747435048104745653
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.