Dirichlet series
| L(s) = 1 | + 2·2-s + 3-s + 4-s − 3·5-s + 2·6-s − 16·7-s + 3·9-s − 6·10-s + 3·11-s + 12-s − 32·14-s − 3·15-s − 2·17-s + 6·18-s + 3·19-s − 3·20-s − 16·21-s + 6·22-s + 15·23-s + 19·25-s − 2·27-s − 16·28-s + 13·29-s − 6·30-s − 11·31-s − 2·32-s + 3·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 6.04·7-s + 9-s − 1.89·10-s + 0.904·11-s + 0.288·12-s − 8.55·14-s − 0.774·15-s − 0.485·17-s + 1.41·18-s + 0.688·19-s − 0.670·20-s − 3.49·21-s + 1.27·22-s + 3.12·23-s + 19/5·25-s − 0.384·27-s − 3.02·28-s + 2.41·29-s − 1.09·30-s − 1.97·31-s − 0.353·32-s + 0.522·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00360869\) |
| Root analytic conductor: | \(0.703613\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4375676941\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4375676941\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 + 11 T + 120 T^{2} + 799 T^{3} + 5429 T^{4} + 799 p T^{5} + 120 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - T - 2 T^{2} + 7 T^{3} - 4 p T^{4} + 2 T^{5} + 16 T^{6} - 44 T^{7} + 73 T^{8} - 44 p T^{9} + 16 p^{2} T^{10} + 2 p^{3} T^{11} - 4 p^{5} T^{12} + 7 p^{5} T^{13} - 2 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 3 T - 2 p T^{2} - 27 T^{3} + 94 T^{4} + 162 T^{5} - 119 p T^{6} - 258 T^{7} + 3661 T^{8} - 258 p T^{9} - 119 p^{3} T^{10} + 162 p^{3} T^{11} + 94 p^{4} T^{12} - 27 p^{5} T^{13} - 2 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 16 T + 127 T^{2} + 95 p T^{3} + 380 p T^{4} + 9169 T^{5} + 29786 T^{6} + 1880 p^{2} T^{7} + 259039 T^{8} + 1880 p^{3} T^{9} + 29786 p^{2} T^{10} + 9169 p^{3} T^{11} + 380 p^{5} T^{12} + 95 p^{6} T^{13} + 127 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 3 T + T^{2} - 12 T^{3} + 36 T^{4} + 261 T^{5} - 586 T^{6} + 3414 T^{7} - 22153 T^{8} + 3414 p T^{9} - 586 p^{2} T^{10} + 261 p^{3} T^{11} + 36 p^{4} T^{12} - 12 p^{5} T^{13} + p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 27 T^{2} + 50 T^{3} + 480 T^{4} - 1000 T^{5} - 4157 T^{6} + 9000 T^{7} + 47779 T^{8} + 9000 p T^{9} - 4157 p^{2} T^{10} - 1000 p^{3} T^{11} + 480 p^{4} T^{12} + 50 p^{5} T^{13} - 27 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T - 3 T^{2} + 9 T^{3} + 128 T^{4} + 981 T^{5} + 82 p T^{6} + 2132 T^{7} + 112113 T^{8} + 2132 p T^{9} + 82 p^{3} T^{10} + 981 p^{3} T^{11} + 128 p^{4} T^{12} + 9 p^{5} T^{13} - 3 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 3 T + 64 T^{2} - 279 T^{3} + 2202 T^{4} - 11568 T^{5} + 53798 T^{6} - 308064 T^{7} + 1089161 T^{8} - 308064 p T^{9} + 53798 p^{2} T^{10} - 11568 p^{3} T^{11} + 2202 p^{4} T^{12} - 279 p^{5} T^{13} + 64 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 15 T + 79 T^{2} - 60 T^{3} - 633 T^{4} - 3810 T^{5} + 36242 T^{6} + 75525 T^{7} - 1303075 T^{8} + 75525 p T^{9} + 36242 p^{2} T^{10} - 3810 p^{3} T^{11} - 633 p^{4} T^{12} - 60 p^{5} T^{13} + 79 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 13 T + 30 T^{2} + 300 T^{3} - 1390 T^{4} - 7929 T^{5} + 57662 T^{6} + 293690 T^{7} - 3872925 T^{8} + 293690 p T^{9} + 57662 p^{2} T^{10} - 7929 p^{3} T^{11} - 1390 p^{4} T^{12} + 300 p^{5} T^{13} + 30 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T - 78 T^{2} + 554 T^{3} + 5603 T^{4} - 26439 T^{5} - 6893 p T^{6} + 327007 T^{7} + 11441583 T^{8} + 327007 p T^{9} - 6893 p^{3} T^{10} - 26439 p^{3} T^{11} + 5603 p^{4} T^{12} + 554 p^{5} T^{13} - 78 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 13 T + 6 T^{2} + 183 T^{3} + 2231 T^{4} + 2181 T^{5} - 186826 T^{6} + 468209 T^{7} + 1750392 T^{8} + 468209 p T^{9} - 186826 p^{2} T^{10} + 2181 p^{3} T^{11} + 2231 p^{4} T^{12} + 183 p^{5} T^{13} + 6 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 73 T^{2} + 420 T^{3} + 4260 T^{4} + 29865 T^{5} + 245708 T^{6} + 2033505 T^{7} + 9196889 T^{8} + 2033505 p T^{9} + 245708 p^{2} T^{10} + 29865 p^{3} T^{11} + 4260 p^{4} T^{12} + 420 p^{5} T^{13} + 73 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 9 T - 82 T^{2} + 897 T^{3} + 18 p T^{4} - 48114 T^{5} + 392170 T^{6} + 968706 T^{7} - 29959051 T^{8} + 968706 p T^{9} + 392170 p^{2} T^{10} - 48114 p^{3} T^{11} + 18 p^{5} T^{12} + 897 p^{5} T^{13} - 82 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 51 T + 1333 T^{2} + 23358 T^{3} + 305973 T^{4} + 3190833 T^{5} + 27951071 T^{6} + 218219904 T^{7} + 1611432353 T^{8} + 218219904 p T^{9} + 27951071 p^{2} T^{10} + 3190833 p^{3} T^{11} + 305973 p^{4} T^{12} + 23358 p^{5} T^{13} + 1333 p^{6} T^{14} + 51 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 18 T + 434 T^{2} + 6189 T^{3} + 85872 T^{4} + 974688 T^{5} + 9899353 T^{6} + 90561729 T^{7} + 724594301 T^{8} + 90561729 p T^{9} + 9899353 p^{2} T^{10} + 974688 p^{3} T^{11} + 85872 p^{4} T^{12} + 6189 p^{5} T^{13} + 434 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 67 | \( 1 + 18 T - 28 T^{2} - 954 T^{3} + 20113 T^{4} + 152739 T^{5} - 1389301 T^{6} + 1098423 T^{7} + 186163333 T^{8} + 1098423 p T^{9} - 1389301 p^{2} T^{10} + 152739 p^{3} T^{11} + 20113 p^{4} T^{12} - 954 p^{5} T^{13} - 28 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 7 T + 66 T^{2} + 1092 T^{3} - 8389 T^{4} + 122679 T^{5} + 285629 T^{6} - 2628694 T^{7} + 99216147 T^{8} - 2628694 p T^{9} + 285629 p^{2} T^{10} + 122679 p^{3} T^{11} - 8389 p^{4} T^{12} + 1092 p^{5} T^{13} + 66 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 13 T + 88 T^{2} + 380 T^{3} - 655 T^{4} + 2272 T^{5} + 334724 T^{6} + 4833155 T^{7} + 47712949 T^{8} + 4833155 p T^{9} + 334724 p^{2} T^{10} + 2272 p^{3} T^{11} - 655 p^{4} T^{12} + 380 p^{5} T^{13} + 88 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 9 T - 146 T^{2} + 1086 T^{3} + 13821 T^{4} - 24237 T^{5} - 1649569 T^{6} - 1358472 T^{7} + 173053847 T^{8} - 1358472 p T^{9} - 1649569 p^{2} T^{10} - 24237 p^{3} T^{11} + 13821 p^{4} T^{12} + 1086 p^{5} T^{13} - 146 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 6 T + 128 T^{2} - 483 T^{3} + 7158 T^{4} + 32232 T^{5} + 54781 T^{6} + 6441891 T^{7} - 233647 T^{8} + 6441891 p T^{9} + 54781 p^{2} T^{10} + 32232 p^{3} T^{11} + 7158 p^{4} T^{12} - 483 p^{5} T^{13} + 128 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 28 T + 435 T^{2} - 5355 T^{3} + 60455 T^{4} - 536409 T^{5} + 3669257 T^{6} - 27140740 T^{7} + 255213705 T^{8} - 27140740 p T^{9} + 3669257 p^{2} T^{10} - 536409 p^{3} T^{11} + 60455 p^{4} T^{12} - 5355 p^{5} T^{13} + 435 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - T + 88 T^{2} + 1093 T^{3} + 17636 T^{4} + 161624 T^{5} + 1223990 T^{6} + 26988604 T^{7} + 155964619 T^{8} + 26988604 p T^{9} + 1223990 p^{2} T^{10} + 161624 p^{3} T^{11} + 17636 p^{4} T^{12} + 1093 p^{5} T^{13} + 88 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.16403253739256490865076095615, −6.97543967239051142869272881752, −6.89766858895365719885270738559, −6.56605909336514256392344611365, −6.42173401092589782873751452400, −6.36505637056999274971184602546, −6.28389632748888373006155382629, −6.23124071506102780257207945864, −6.10113108237928422046358807716, −5.64233122636620560125327929549, −5.28566759733401299641919198799, −4.93257207526502555921603186419, −4.91247215419825610432431463611, −4.69821221689604822434359648625, −4.48597119978132507112015533799, −4.17870059315614226616791842113, −4.10944466169911802392665271889, −3.57883045726904586046247729707, −3.48450706379823773720029534292, −3.25722867155688086807073171632, −3.22923032353409435971154435935, −2.94079173400474424739659266950, −2.78168076086641373317387597992, −2.75713476615165345604856051944, −1.28426446530966460200095964470, 1.28426446530966460200095964470, 2.75713476615165345604856051944, 2.78168076086641373317387597992, 2.94079173400474424739659266950, 3.22923032353409435971154435935, 3.25722867155688086807073171632, 3.48450706379823773720029534292, 3.57883045726904586046247729707, 4.10944466169911802392665271889, 4.17870059315614226616791842113, 4.48597119978132507112015533799, 4.69821221689604822434359648625, 4.91247215419825610432431463611, 4.93257207526502555921603186419, 5.28566759733401299641919198799, 5.64233122636620560125327929549, 6.10113108237928422046358807716, 6.23124071506102780257207945864, 6.28389632748888373006155382629, 6.36505637056999274971184602546, 6.42173401092589782873751452400, 6.56605909336514256392344611365, 6.89766858895365719885270738559, 6.97543967239051142869272881752, 7.16403253739256490865076095615
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.