Dirichlet series
| L(s) = 1 | + 5·2-s + 54·4-s + 42·5-s + 748·7-s + 593·8-s + 210·10-s + 1.07e3·11-s + 3.74e3·14-s + 6.97e3·16-s − 7.21e3·17-s − 2.46e4·19-s + 2.26e3·20-s + 5.35e3·22-s + 1.52e4·23-s − 1.92e4·25-s + 4.03e4·28-s − 3.25e4·29-s + 4.02e4·31-s + 3.24e4·32-s − 3.60e4·34-s + 3.14e4·35-s − 4.56e4·37-s − 1.23e5·38-s + 2.49e4·40-s − 4.45e5·43-s + 5.77e4·44-s + 7.61e4·46-s + ⋯ |
| L(s) = 1 | + 5/8·2-s + 0.843·4-s + 0.335·5-s + 2.18·7-s + 1.15·8-s + 0.209·10-s + 0.803·11-s + 1.36·14-s + 1.70·16-s − 1.46·17-s − 3.58·19-s + 0.283·20-s + 0.502·22-s + 1.25·23-s − 1.23·25-s + 1.84·28-s − 1.33·29-s + 1.34·31-s + 0.991·32-s − 0.917·34-s + 0.732·35-s − 0.901·37-s − 2.24·38-s + 0.389·40-s − 5.60·43-s + 0.678·44-s + 0.782·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.94699\times 10^{9}\) |
| Root analytic conductor: | \(3.80702\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(4.311051112\) |
| \(L(\frac12)\) | \(\approx\) | \(4.311051112\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 748 T + 38242 p T^{2} - 125432 p^{3} T^{3} + 489929 p^{5} T^{4} - 125432 p^{9} T^{5} + 38242 p^{13} T^{6} - 748 p^{18} T^{7} + p^{24} T^{8} \) | |
| good | 2 | \( 1 - 5 T - 29 T^{2} - 89 p T^{3} - 777 p T^{4} + 9243 p^{2} T^{5} + 50201 p^{2} T^{6} - 71679 p^{5} T^{7} + 17 p^{17} T^{8} - 71679 p^{11} T^{9} + 50201 p^{14} T^{10} + 9243 p^{20} T^{11} - 777 p^{25} T^{12} - 89 p^{31} T^{13} - 29 p^{36} T^{14} - 5 p^{42} T^{15} + p^{48} T^{16} \) |
| 5 | \( 1 - 42 T + 4199 p T^{2} - 857094 T^{3} + 59255749 T^{4} - 71357004 p^{3} T^{5} - 55623489398 p^{2} T^{6} - 110556133392 p^{5} T^{7} - 1151873646254 p^{4} T^{8} - 110556133392 p^{11} T^{9} - 55623489398 p^{14} T^{10} - 71357004 p^{21} T^{11} + 59255749 p^{24} T^{12} - 857094 p^{30} T^{13} + 4199 p^{37} T^{14} - 42 p^{42} T^{15} + p^{48} T^{16} \) | |
| 11 | \( 1 - 1070 T - 516103 p T^{2} + 3448075790 T^{3} + 22836612400425 T^{4} - 7759923983215740 T^{5} - 60967102988727183826 T^{6} + \)\(45\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!86\)\( T^{8} + \)\(45\!\cdots\!20\)\( p^{6} T^{9} - 60967102988727183826 p^{12} T^{10} - 7759923983215740 p^{18} T^{11} + 22836612400425 p^{24} T^{12} + 3448075790 p^{30} T^{13} - 516103 p^{37} T^{14} - 1070 p^{42} T^{15} + p^{48} T^{16} \) | |
| 13 | \( 1 - 16298786 T^{2} + 111852524426401 T^{4} - 37539779661680550026 p T^{6} + \)\(20\!\cdots\!88\)\( T^{8} - 37539779661680550026 p^{13} T^{10} + 111852524426401 p^{24} T^{12} - 16298786 p^{36} T^{14} + p^{48} T^{16} \) | |
| 17 | \( 1 + 7212 T + 37951384 T^{2} + 148666264032 T^{3} - 401372098722782 T^{4} - 760743811350012996 T^{5} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!47\)\( T^{8} + \)\(10\!\cdots\!52\)\( p^{6} T^{9} + \)\(67\!\cdots\!80\)\( p^{12} T^{10} - 760743811350012996 p^{18} T^{11} - 401372098722782 p^{24} T^{12} + 148666264032 p^{30} T^{13} + 37951384 p^{36} T^{14} + 7212 p^{42} T^{15} + p^{48} T^{16} \) | |
| 19 | \( 1 + 24606 T + 362733151 T^{2} + 3959468067834 T^{3} + 36560849427203149 T^{4} + \)\(28\!\cdots\!40\)\( T^{5} + \)\(19\!\cdots\!30\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(82\!\cdots\!38\)\( T^{8} + \)\(12\!\cdots\!40\)\( p^{6} T^{9} + \)\(19\!\cdots\!30\)\( p^{12} T^{10} + \)\(28\!\cdots\!40\)\( p^{18} T^{11} + 36560849427203149 p^{24} T^{12} + 3959468067834 p^{30} T^{13} + 362733151 p^{36} T^{14} + 24606 p^{42} T^{15} + p^{48} T^{16} \) | |
| 23 | \( 1 - 15224 T - 71404700 T^{2} + 3251082330032 T^{3} - 39028069690313382 T^{4} + \)\(38\!\cdots\!08\)\( T^{5} - \)\(16\!\cdots\!68\)\( T^{6} - \)\(76\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!71\)\( T^{8} - \)\(76\!\cdots\!84\)\( p^{6} T^{9} - \)\(16\!\cdots\!68\)\( p^{12} T^{10} + \)\(38\!\cdots\!08\)\( p^{18} T^{11} - 39028069690313382 p^{24} T^{12} + 3251082330032 p^{30} T^{13} - 71404700 p^{36} T^{14} - 15224 p^{42} T^{15} + p^{48} T^{16} \) | |
| 29 | \( ( 1 + 16262 T + 1936712745 T^{2} + 19588912728706 T^{3} + 1557131395176891808 T^{4} + 19588912728706 p^{6} T^{5} + 1936712745 p^{12} T^{6} + 16262 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 31 | \( 1 - 40200 T + 2343974770 T^{2} - 72572849754000 T^{3} + 2084077897304297293 T^{4} - \)\(75\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!50\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} + \)\(32\!\cdots\!28\)\( T^{8} - \)\(10\!\cdots\!60\)\( p^{6} T^{9} + \)\(25\!\cdots\!50\)\( p^{12} T^{10} - \)\(75\!\cdots\!20\)\( p^{18} T^{11} + 2084077897304297293 p^{24} T^{12} - 72572849754000 p^{30} T^{13} + 2343974770 p^{36} T^{14} - 40200 p^{42} T^{15} + p^{48} T^{16} \) | |
| 37 | \( 1 + 45670 T - 5087844981 T^{2} - 327542380882710 T^{3} + 12351146188805096265 T^{4} + \)\(98\!\cdots\!60\)\( T^{5} - \)\(98\!\cdots\!54\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} - \)\(38\!\cdots\!86\)\( T^{8} - \)\(12\!\cdots\!80\)\( p^{6} T^{9} - \)\(98\!\cdots\!54\)\( p^{12} T^{10} + \)\(98\!\cdots\!60\)\( p^{18} T^{11} + 12351146188805096265 p^{24} T^{12} - 327542380882710 p^{30} T^{13} - 5087844981 p^{36} T^{14} + 45670 p^{42} T^{15} + p^{48} T^{16} \) | |
| 41 | \( 1 - 26005847576 T^{2} + \)\(34\!\cdots\!60\)\( T^{4} - \)\(28\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!64\)\( p^{12} T^{10} + \)\(34\!\cdots\!60\)\( p^{24} T^{12} - 26005847576 p^{36} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 + 222830 T + 23708395369 T^{2} + 1380667882433470 T^{3} + 73296626267097035876 T^{4} + 1380667882433470 p^{6} T^{5} + 23708395369 p^{12} T^{6} + 222830 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( 1 + 82884 T + 33030950488 T^{2} + 2547939641253024 T^{3} + \)\(61\!\cdots\!34\)\( T^{4} + \)\(67\!\cdots\!36\)\( T^{5} + \)\(88\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} + \)\(95\!\cdots\!35\)\( T^{8} + \)\(10\!\cdots\!28\)\( p^{6} T^{9} + \)\(88\!\cdots\!56\)\( p^{12} T^{10} + \)\(67\!\cdots\!36\)\( p^{18} T^{11} + \)\(61\!\cdots\!34\)\( p^{24} T^{12} + 2547939641253024 p^{30} T^{13} + 33030950488 p^{36} T^{14} + 82884 p^{42} T^{15} + p^{48} T^{16} \) | |
| 53 | \( 1 - 13034 T - 37911011717 T^{2} - 368806876088782 T^{3} + \)\(25\!\cdots\!77\)\( T^{4} + \)\(27\!\cdots\!36\)\( T^{5} - \)\(79\!\cdots\!10\)\( T^{6} - \)\(35\!\cdots\!88\)\( T^{7} + \)\(46\!\cdots\!02\)\( T^{8} - \)\(35\!\cdots\!88\)\( p^{6} T^{9} - \)\(79\!\cdots\!10\)\( p^{12} T^{10} + \)\(27\!\cdots\!36\)\( p^{18} T^{11} + \)\(25\!\cdots\!77\)\( p^{24} T^{12} - 368806876088782 p^{30} T^{13} - 37911011717 p^{36} T^{14} - 13034 p^{42} T^{15} + p^{48} T^{16} \) | |
| 59 | \( 1 + 1810362 T + 1656169621891 T^{2} + 1020500030287048566 T^{3} + \)\(47\!\cdots\!61\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(54\!\cdots\!70\)\( T^{6} + \)\(14\!\cdots\!12\)\( T^{7} + \)\(31\!\cdots\!62\)\( T^{8} + \)\(14\!\cdots\!12\)\( p^{6} T^{9} + \)\(54\!\cdots\!70\)\( p^{12} T^{10} + \)\(17\!\cdots\!16\)\( p^{18} T^{11} + \)\(47\!\cdots\!61\)\( p^{24} T^{12} + 1020500030287048566 p^{30} T^{13} + 1656169621891 p^{36} T^{14} + 1810362 p^{42} T^{15} + p^{48} T^{16} \) | |
| 61 | \( 1 + 392856 T + 253766894356 T^{2} + 79483260556868064 T^{3} + \)\(32\!\cdots\!58\)\( T^{4} + \)\(85\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(61\!\cdots\!24\)\( T^{7} + \)\(16\!\cdots\!47\)\( T^{8} + \)\(61\!\cdots\!24\)\( p^{6} T^{9} + \)\(27\!\cdots\!00\)\( p^{12} T^{10} + \)\(85\!\cdots\!52\)\( p^{18} T^{11} + \)\(32\!\cdots\!58\)\( p^{24} T^{12} + 79483260556868064 p^{30} T^{13} + 253766894356 p^{36} T^{14} + 392856 p^{42} T^{15} + p^{48} T^{16} \) | |
| 67 | \( 1 - 384094 T - 169408100025 T^{2} + 57304941612054286 T^{3} + \)\(24\!\cdots\!05\)\( T^{4} - \)\(42\!\cdots\!12\)\( T^{5} - \)\(32\!\cdots\!54\)\( T^{6} + \)\(98\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!22\)\( T^{8} + \)\(98\!\cdots\!00\)\( p^{6} T^{9} - \)\(32\!\cdots\!54\)\( p^{12} T^{10} - \)\(42\!\cdots\!12\)\( p^{18} T^{11} + \)\(24\!\cdots\!05\)\( p^{24} T^{12} + 57304941612054286 p^{30} T^{13} - 169408100025 p^{36} T^{14} - 384094 p^{42} T^{15} + p^{48} T^{16} \) | |
| 71 | \( ( 1 + 112844 T + 324448495296 T^{2} - 1073384954931548 T^{3} + \)\(47\!\cdots\!38\)\( T^{4} - 1073384954931548 p^{6} T^{5} + 324448495296 p^{12} T^{6} + 112844 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 73 | \( 1 - 903078 T + 897300437755 T^{2} - 564830568330899706 T^{3} + \)\(36\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!76\)\( T^{5} + \)\(93\!\cdots\!70\)\( T^{6} - \)\(39\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(39\!\cdots\!56\)\( p^{6} T^{9} + \)\(93\!\cdots\!70\)\( p^{12} T^{10} - \)\(18\!\cdots\!76\)\( p^{18} T^{11} + \)\(36\!\cdots\!49\)\( p^{24} T^{12} - 564830568330899706 p^{30} T^{13} + 897300437755 p^{36} T^{14} - 903078 p^{42} T^{15} + p^{48} T^{16} \) | |
| 79 | \( 1 + 559592 T - 192861426366 T^{2} - 38884411164730496 T^{3} + \)\(45\!\cdots\!81\)\( T^{4} - \)\(13\!\cdots\!84\)\( T^{5} + \)\(30\!\cdots\!58\)\( T^{6} + \)\(45\!\cdots\!96\)\( T^{7} - \)\(14\!\cdots\!88\)\( T^{8} + \)\(45\!\cdots\!96\)\( p^{6} T^{9} + \)\(30\!\cdots\!58\)\( p^{12} T^{10} - \)\(13\!\cdots\!84\)\( p^{18} T^{11} + \)\(45\!\cdots\!81\)\( p^{24} T^{12} - 38884411164730496 p^{30} T^{13} - 192861426366 p^{36} T^{14} + 559592 p^{42} T^{15} + p^{48} T^{16} \) | |
| 83 | \( 1 - 577456669706 T^{2} + \)\(27\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(42\!\cdots\!16\)\( T^{8} - \)\(12\!\cdots\!18\)\( p^{12} T^{10} + \)\(27\!\cdots\!89\)\( p^{24} T^{12} - 577456669706 p^{36} T^{14} + p^{48} T^{16} \) | |
| 89 | \( 1 - 1770036 T + 3412957024480 T^{2} - 4192533013088545728 T^{3} + \)\(53\!\cdots\!74\)\( T^{4} - \)\(51\!\cdots\!96\)\( T^{5} + \)\(49\!\cdots\!64\)\( T^{6} - \)\(38\!\cdots\!04\)\( T^{7} + \)\(30\!\cdots\!35\)\( T^{8} - \)\(38\!\cdots\!04\)\( p^{6} T^{9} + \)\(49\!\cdots\!64\)\( p^{12} T^{10} - \)\(51\!\cdots\!96\)\( p^{18} T^{11} + \)\(53\!\cdots\!74\)\( p^{24} T^{12} - 4192533013088545728 p^{30} T^{13} + 3412957024480 p^{36} T^{14} - 1770036 p^{42} T^{15} + p^{48} T^{16} \) | |
| 97 | \( 1 - 4315064900690 T^{2} + \)\(91\!\cdots\!41\)\( T^{4} - \)\(12\!\cdots\!58\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{8} - \)\(12\!\cdots\!58\)\( p^{12} T^{10} + \)\(91\!\cdots\!41\)\( p^{24} T^{12} - 4315064900690 p^{36} T^{14} + p^{48} T^{16} \) | |
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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
−5.65905608152725433705234570161, −5.57235468914578000483016081330, −5.24365568215911412198672999075, −5.20887882079749377586893256440, −4.77278383967155538711240447239, −4.59063637629914631976030774994, −4.57764890793342583966692050424, −4.55371846163410223625969655702, −4.37515496717034008978944031031, −4.19452207920093372111786031120, −3.72215772702598021602376049643, −3.42891405291289229644577464113, −3.24386794146497342135078290495, −3.10630892935579483322615554115, −3.06375428261589047290131672149, −2.51525542642418265646446098610, −2.01703089857938073765570228690, −1.82362564777650851639315442156, −1.80406698126303704139057241160, −1.76789983486328433699418765989, −1.53681210692477839869878743345, −1.51929787249147830729549186108, −0.57306509378343003738053809781, −0.56215191068158929534357809720, −0.13976195746827222360761963914, 0.13976195746827222360761963914, 0.56215191068158929534357809720, 0.57306509378343003738053809781, 1.51929787249147830729549186108, 1.53681210692477839869878743345, 1.76789983486328433699418765989, 1.80406698126303704139057241160, 1.82362564777650851639315442156, 2.01703089857938073765570228690, 2.51525542642418265646446098610, 3.06375428261589047290131672149, 3.10630892935579483322615554115, 3.24386794146497342135078290495, 3.42891405291289229644577464113, 3.72215772702598021602376049643, 4.19452207920093372111786031120, 4.37515496717034008978944031031, 4.55371846163410223625969655702, 4.57764890793342583966692050424, 4.59063637629914631976030774994, 4.77278383967155538711240447239, 5.20887882079749377586893256440, 5.24365568215911412198672999075, 5.57235468914578000483016081330, 5.65905608152725433705234570161
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.