Dirichlet series
| L(s) = 1 | + 8·2-s + 24·4-s − 4·5-s + 25·7-s − 32·10-s − 56·11-s − 18·13-s + 200·14-s − 240·16-s + 118·17-s + 37·19-s − 96·20-s − 448·22-s − 200·23-s + 206·25-s − 144·26-s + 600·28-s + 524·29-s + 276·31-s − 768·32-s + 944·34-s − 100·35-s − 185·37-s + 296·38-s − 60·41-s − 1.55e3·43-s − 1.34e3·44-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s − 0.357·5-s + 1.34·7-s − 1.01·10-s − 1.53·11-s − 0.384·13-s + 3.81·14-s − 3.75·16-s + 1.68·17-s + 0.446·19-s − 1.07·20-s − 4.34·22-s − 1.81·23-s + 1.64·25-s − 1.08·26-s + 4.04·28-s + 3.35·29-s + 1.59·31-s − 4.24·32-s + 4.76·34-s − 0.482·35-s − 0.821·37-s + 1.26·38-s − 0.228·41-s − 5.51·43-s − 4.60·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{24} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.12157\times 10^{10}\) |
| Root analytic conductor: | \(4.72257\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(2.601141517\) |
| \(L(\frac12)\) | \(\approx\) | \(2.601141517\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 25 T + 892 T^{2} - 257 p^{2} T^{3} + 986 p^{3} T^{4} - 257 p^{5} T^{5} + 892 p^{6} T^{6} - 25 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 5 | \( 1 + 4 T - 38 p T^{2} - 952 T^{3} + 3122 p T^{4} + 107736 T^{5} + 79504 p^{2} T^{6} - 4377188 T^{7} - 401592049 T^{8} - 4377188 p^{3} T^{9} + 79504 p^{8} T^{10} + 107736 p^{9} T^{11} + 3122 p^{13} T^{12} - 952 p^{15} T^{13} - 38 p^{19} T^{14} + 4 p^{21} T^{15} + p^{24} T^{16} \) |
| 11 | \( 1 + 56 T - 1294 T^{2} - 27848 T^{3} + 4620514 T^{4} - 20370420 T^{5} - 7796509808 T^{6} - 51271167136 T^{7} + 4524706836959 T^{8} - 51271167136 p^{3} T^{9} - 7796509808 p^{6} T^{10} - 20370420 p^{9} T^{11} + 4620514 p^{12} T^{12} - 27848 p^{15} T^{13} - 1294 p^{18} T^{14} + 56 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( ( 1 + 9 T + 6151 T^{2} + 57820 T^{3} + 18686898 T^{4} + 57820 p^{3} T^{5} + 6151 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 17 | \( 1 - 118 T - 4258 T^{2} + 842176 T^{3} + 26902678 T^{4} - 4430920662 T^{5} - 73353446888 T^{6} + 10771697769914 T^{7} + 10410332361635 T^{8} + 10771697769914 p^{3} T^{9} - 73353446888 p^{6} T^{10} - 4430920662 p^{9} T^{11} + 26902678 p^{12} T^{12} + 842176 p^{15} T^{13} - 4258 p^{18} T^{14} - 118 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 37 T - 4167 T^{2} + 433714 T^{3} - 61048363 T^{4} + 1953920457 T^{5} + 99571342114 T^{6} - 22833789745645 T^{7} + 2910424159827918 T^{8} - 22833789745645 p^{3} T^{9} + 99571342114 p^{6} T^{10} + 1953920457 p^{9} T^{11} - 61048363 p^{12} T^{12} + 433714 p^{15} T^{13} - 4167 p^{18} T^{14} - 37 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 200 T + 19268 T^{2} + 4502992 T^{3} + 578747002 T^{4} + 39443031096 T^{5} + 8952169699408 T^{6} + 47138759811832 p T^{7} + 70993179469128755 T^{8} + 47138759811832 p^{4} T^{9} + 8952169699408 p^{6} T^{10} + 39443031096 p^{9} T^{11} + 578747002 p^{12} T^{12} + 4502992 p^{15} T^{13} + 19268 p^{18} T^{14} + 200 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( ( 1 - 262 T + 70382 T^{2} - 7734354 T^{3} + 1506015426 T^{4} - 7734354 p^{3} T^{5} + 70382 p^{6} T^{6} - 262 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 31 | \( 1 - 276 T + 25820 T^{2} - 10857088 T^{3} + 1976132059 T^{4} - 44637997712 T^{5} + 72146240415876 T^{6} - 15282614101089316 T^{7} + 940693478637184968 T^{8} - 15282614101089316 p^{3} T^{9} + 72146240415876 p^{6} T^{10} - 44637997712 p^{9} T^{11} + 1976132059 p^{12} T^{12} - 10857088 p^{15} T^{13} + 25820 p^{18} T^{14} - 276 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 5 p T - 28182 T^{2} - 19254761 T^{3} - 5339642509 T^{4} - 621423096180 T^{5} + 14603173107124 T^{6} + 54636230423573108 T^{7} + 21079140975815813460 T^{8} + 54636230423573108 p^{3} T^{9} + 14603173107124 p^{6} T^{10} - 621423096180 p^{9} T^{11} - 5339642509 p^{12} T^{12} - 19254761 p^{15} T^{13} - 28182 p^{18} T^{14} + 5 p^{22} T^{15} + p^{24} T^{16} \) | |
| 41 | \( ( 1 + 30 T - 53770 T^{2} + 10546866 T^{3} + 6199463634 T^{4} + 10546866 p^{3} T^{5} - 53770 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 43 | \( ( 1 + 778 T + 455356 T^{2} + 180370816 T^{3} + 59455969525 T^{4} + 180370816 p^{3} T^{5} + 455356 p^{6} T^{6} + 778 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 + 30 T - 233978 T^{2} - 7042728 T^{3} + 20874591478 T^{4} + 341085867174 T^{5} - 2913767666766488 T^{6} + 6045629293426602 T^{7} + \)\(44\!\cdots\!31\)\( T^{8} + 6045629293426602 p^{3} T^{9} - 2913767666766488 p^{6} T^{10} + 341085867174 p^{9} T^{11} + 20874591478 p^{12} T^{12} - 7042728 p^{15} T^{13} - 233978 p^{18} T^{14} + 30 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 480 T - 133052 T^{2} - 78547008 T^{3} + 2880222202 T^{4} - 945699549984 T^{5} - 1820912761525232 T^{6} + 425561990543160096 T^{7} + \)\(41\!\cdots\!55\)\( T^{8} + 425561990543160096 p^{3} T^{9} - 1820912761525232 p^{6} T^{10} - 945699549984 p^{9} T^{11} + 2880222202 p^{12} T^{12} - 78547008 p^{15} T^{13} - 133052 p^{18} T^{14} + 480 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 296 T - 225094 T^{2} - 26071576 T^{3} + 48047416978 T^{4} + 6396496161492 T^{5} + 12987765766551856 T^{6} - 3524743554415297424 T^{7} - \)\(30\!\cdots\!89\)\( T^{8} - 3524743554415297424 p^{3} T^{9} + 12987765766551856 p^{6} T^{10} + 6396496161492 p^{9} T^{11} + 48047416978 p^{12} T^{12} - 26071576 p^{15} T^{13} - 225094 p^{18} T^{14} - 296 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 474 T - 78214 T^{2} - 300891220 T^{3} + 155813021293 T^{4} + 16287352656676 T^{5} + 53973600563787870 T^{6} - 25862651144467518334 T^{7} - \)\(28\!\cdots\!60\)\( T^{8} - 25862651144467518334 p^{3} T^{9} + 53973600563787870 p^{6} T^{10} + 16287352656676 p^{9} T^{11} + 155813021293 p^{12} T^{12} - 300891220 p^{15} T^{13} - 78214 p^{18} T^{14} - 474 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 1319 T + 435894 T^{2} + 260547003 T^{3} - 228312863079 T^{4} + 52877715626208 T^{5} + 12075026282780822 T^{6} - 31080208875847270882 T^{7} + \)\(25\!\cdots\!76\)\( T^{8} - 31080208875847270882 p^{3} T^{9} + 12075026282780822 p^{6} T^{10} + 52877715626208 p^{9} T^{11} - 228312863079 p^{12} T^{12} + 260547003 p^{15} T^{13} + 435894 p^{18} T^{14} - 1319 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( ( 1 - 926 T + 1215866 T^{2} - 651674910 T^{3} + 558490511946 T^{4} - 651674910 p^{3} T^{5} + 1215866 p^{6} T^{6} - 926 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 73 | \( 1 + 1423 T + 151179 T^{2} - 356462016 T^{3} + 225594950031 T^{4} + 190838170992351 T^{5} - 111189108199360714 T^{6} - 1240152686167800121 T^{7} + \)\(82\!\cdots\!20\)\( T^{8} - 1240152686167800121 p^{3} T^{9} - 111189108199360714 p^{6} T^{10} + 190838170992351 p^{9} T^{11} + 225594950031 p^{12} T^{12} - 356462016 p^{15} T^{13} + 151179 p^{18} T^{14} + 1423 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 765 T - 740524 T^{2} + 287899523 T^{3} + 440031712255 T^{4} + 69755733771928 T^{5} - 263635131983182680 T^{6} - 20666144138306564776 T^{7} + \)\(11\!\cdots\!60\)\( T^{8} - 20666144138306564776 p^{3} T^{9} - 263635131983182680 p^{6} T^{10} + 69755733771928 p^{9} T^{11} + 440031712255 p^{12} T^{12} + 287899523 p^{15} T^{13} - 740524 p^{18} T^{14} - 765 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( ( 1 - 10 p T + 2276504 T^{2} - 1415613270 T^{3} + 1949801366142 T^{4} - 1415613270 p^{3} T^{5} + 2276504 p^{6} T^{6} - 10 p^{10} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 - 864 T - 2003126 T^{2} + 915016392 T^{3} + 36952363898 p T^{4} - 759050670074364 T^{5} - 3398637459395982080 T^{6} + \)\(20\!\cdots\!36\)\( T^{7} + \)\(27\!\cdots\!27\)\( T^{8} + \)\(20\!\cdots\!36\)\( p^{3} T^{9} - 3398637459395982080 p^{6} T^{10} - 759050670074364 p^{9} T^{11} + 36952363898 p^{13} T^{12} + 915016392 p^{15} T^{13} - 2003126 p^{18} T^{14} - 864 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( ( 1 - 544 T + 2676994 T^{2} - 857057120 T^{3} + 3128962808483 T^{4} - 857057120 p^{3} T^{5} + 2676994 p^{6} T^{6} - 544 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
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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
−4.77087438290865956155777571270, −4.69460247562870259117520781086, −4.12236570069737637796489409379, −4.09316699384181222764124090395, −4.04792877512697482195685866141, −3.83115524516218526794955051737, −3.80220911749361987296092249267, −3.43835058544111494226552412324, −3.36768892054348674353313745940, −3.23407876064048439419595253761, −3.08040547777373522846601859957, −3.04307590418750820041490286876, −2.79511338873265944527880194296, −2.60160759894985429364896137435, −2.52914059314062382503992380566, −2.24534000652016275993402283988, −1.92274628572807976164754840528, −1.80022967933081745604693068960, −1.68911399444701172437930817569, −1.50747775298019524446271504091, −0.923940849618747846577788330444, −0.912976215873091690015010419648, −0.62507684867851037213381026252, −0.57686313330881880613319369787, −0.06000680351829049141800849793, 0.06000680351829049141800849793, 0.57686313330881880613319369787, 0.62507684867851037213381026252, 0.912976215873091690015010419648, 0.923940849618747846577788330444, 1.50747775298019524446271504091, 1.68911399444701172437930817569, 1.80022967933081745604693068960, 1.92274628572807976164754840528, 2.24534000652016275993402283988, 2.52914059314062382503992380566, 2.60160759894985429364896137435, 2.79511338873265944527880194296, 3.04307590418750820041490286876, 3.08040547777373522846601859957, 3.23407876064048439419595253761, 3.36768892054348674353313745940, 3.43835058544111494226552412324, 3.80220911749361987296092249267, 3.83115524516218526794955051737, 4.04792877512697482195685866141, 4.09316699384181222764124090395, 4.12236570069737637796489409379, 4.69460247562870259117520781086, 4.77087438290865956155777571270
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.