Dirichlet series
| L(s) = 1 | + 64·2-s + 324·3-s + 2.56e3·4-s + 942·5-s + 2.07e4·6-s + 5.08e3·7-s + 8.19e4·8-s + 6.56e4·9-s + 6.02e4·10-s − 2.08e4·11-s + 8.29e5·12-s + 2.60e5·13-s + 3.25e5·14-s + 3.05e5·15-s + 2.29e6·16-s + 3.34e5·17-s + 4.19e6·18-s + 1.12e6·19-s + 2.41e6·20-s + 1.64e6·21-s − 1.33e6·22-s + 3.42e5·23-s + 2.65e7·24-s − 1.62e6·25-s + 1.66e7·26-s + 1.06e7·27-s + 1.30e7·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s + 0.674·5-s + 6.53·6-s + 0.800·7-s + 7.07·8-s + 10/3·9-s + 1.90·10-s − 0.428·11-s + 11.5·12-s + 2.53·13-s + 2.26·14-s + 1.55·15-s + 35/4·16-s + 0.970·17-s + 9.42·18-s + 1.98·19-s + 3.37·20-s + 1.84·21-s − 1.21·22-s + 0.255·23-s + 16.3·24-s − 0.832·25-s + 7.16·26-s + 3.84·27-s + 4.00·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 17^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.61641\times 10^{6}\) |
| Root analytic conductor: | \(7.24801\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(432.7199342\) |
| \(L(\frac12)\) | \(\approx\) | \(432.7199342\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 17 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 942 T + 2513177 T^{2} - 199269254 p T^{3} + 248455201404 p^{2} T^{4} - 199269254 p^{10} T^{5} + 2513177 p^{18} T^{6} - 942 p^{27} T^{7} + p^{36} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 5088 T + 10711044 p T^{2} - 1309074336 p^{3} T^{3} + 13042887597898 p^{3} T^{4} - 1309074336 p^{12} T^{5} + 10711044 p^{19} T^{6} - 5088 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 20822 T + 5200446045 T^{2} + 162407960699694 T^{3} + 14735046737655335836 T^{4} + 162407960699694 p^{9} T^{5} + 5200446045 p^{18} T^{6} + 20822 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 260722 T + 52514528265 T^{2} - 44770841630930 p^{2} T^{3} + \)\(87\!\cdots\!44\)\( T^{4} - 44770841630930 p^{11} T^{5} + 52514528265 p^{18} T^{6} - 260722 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 1127114 T + 1305453782845 T^{2} - 938080635866799154 T^{3} + \)\(63\!\cdots\!48\)\( T^{4} - 938080635866799154 p^{9} T^{5} + 1305453782845 p^{18} T^{6} - 1127114 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 342858 T + 2047828581293 T^{2} + 4229256375065294838 T^{3} - \)\(75\!\cdots\!20\)\( T^{4} + 4229256375065294838 p^{9} T^{5} + 2047828581293 p^{18} T^{6} - 342858 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 3453272 T + 16723369193116 T^{2} + 70936160031324843896 T^{3} - \)\(20\!\cdots\!14\)\( T^{4} + 70936160031324843896 p^{9} T^{5} + 16723369193116 p^{18} T^{6} - 3453272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 8734524 T + 47077621271488 T^{2} + 81617446328228058004 T^{3} - \)\(85\!\cdots\!78\)\( T^{4} + 81617446328228058004 p^{9} T^{5} + 47077621271488 p^{18} T^{6} - 8734524 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 4427508 T + 170245936358216 T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(19\!\cdots\!88\)\( p^{9} T^{5} + 170245936358216 p^{18} T^{6} - 4427508 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 12694878 T + 571000448443169 T^{2} + \)\(33\!\cdots\!74\)\( T^{3} + \)\(13\!\cdots\!76\)\( T^{4} + \)\(33\!\cdots\!74\)\( p^{9} T^{5} + 571000448443169 p^{18} T^{6} + 12694878 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 11513590 T + 1919754639109133 T^{2} - \)\(16\!\cdots\!06\)\( T^{3} + \)\(14\!\cdots\!92\)\( T^{4} - \)\(16\!\cdots\!06\)\( p^{9} T^{5} + 1919754639109133 p^{18} T^{6} - 11513590 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 1620412 T + 3186517820480528 T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(45\!\cdots\!74\)\( T^{4} - \)\(12\!\cdots\!88\)\( p^{9} T^{5} + 3186517820480528 p^{18} T^{6} + 1620412 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 42954392 T + 9829335077803292 T^{2} - \)\(46\!\cdots\!56\)\( T^{3} + \)\(42\!\cdots\!98\)\( T^{4} - \)\(46\!\cdots\!56\)\( p^{9} T^{5} + 9829335077803292 p^{18} T^{6} - 42954392 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 30959244 T + 14141345826426368 T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!62\)\( T^{4} + \)\(13\!\cdots\!40\)\( p^{9} T^{5} + 14141345826426368 p^{18} T^{6} + 30959244 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 52264348 T + 16258175211362104 T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} - \)\(10\!\cdots\!96\)\( p^{9} T^{5} + 16258175211362104 p^{18} T^{6} + 52264348 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 268395776 T + 94879502505139948 T^{2} - \)\(13\!\cdots\!32\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!32\)\( p^{9} T^{5} + 94879502505139948 p^{18} T^{6} - 268395776 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 405104640 T + 196082843800580780 T^{2} - \)\(47\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(47\!\cdots\!60\)\( p^{9} T^{5} + 196082843800580780 p^{18} T^{6} - 405104640 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 80652312 T + 220296531505778444 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!08\)\( p^{9} T^{5} + 220296531505778444 p^{18} T^{6} - 80652312 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 153459652 T + 351535263707479936 T^{2} + \)\(43\!\cdots\!68\)\( T^{3} + \)\(59\!\cdots\!66\)\( T^{4} + \)\(43\!\cdots\!68\)\( p^{9} T^{5} + 351535263707479936 p^{18} T^{6} + 153459652 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 963057596 T + 753242391013504368 T^{2} - \)\(43\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!66\)\( T^{4} - \)\(43\!\cdots\!60\)\( p^{9} T^{5} + 753242391013504368 p^{18} T^{6} - 963057596 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1052288804 T + 1511343518252651000 T^{2} - \)\(98\!\cdots\!24\)\( T^{3} + \)\(79\!\cdots\!78\)\( T^{4} - \)\(98\!\cdots\!24\)\( p^{9} T^{5} + 1511343518252651000 p^{18} T^{6} - 1052288804 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 340518220 T + 2842687229221943880 T^{2} - \)\(68\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!62\)\( T^{4} - \)\(68\!\cdots\!08\)\( p^{9} T^{5} + 2842687229221943880 p^{18} T^{6} - 340518220 p^{27} T^{7} + p^{36} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.400597831741162369422740489997, −7.69408136750643354346208589279, −7.66996318468826532431755722387, −7.61053511825945128538483787804, −7.31945320421481453145812182693, −6.43430435568886477839358906226, −6.39344764440031256656743352707, −6.25375694774902370799747778372, −6.07402294226655892280926950871, −5.22757285037087253942056979328, −5.17546941442672627763060288287, −4.88229729551507725124123543426, −4.81364852128904205695527148469, −3.91276176140831411905610703583, −3.84801286158904071336871570042, −3.60400932766711015802880228142, −3.48097423040603140407962548083, −3.04096669046518301647300730150, −2.53588402199264398097523930543, −2.51335848937638530059648101722, −2.15583250461161346530355996139, −1.54613552673476411772990172076, −1.26720096324756830090096338295, −1.15652616571748931842182441642, −0.806235984038850443909744718004, 0.806235984038850443909744718004, 1.15652616571748931842182441642, 1.26720096324756830090096338295, 1.54613552673476411772990172076, 2.15583250461161346530355996139, 2.51335848937638530059648101722, 2.53588402199264398097523930543, 3.04096669046518301647300730150, 3.48097423040603140407962548083, 3.60400932766711015802880228142, 3.84801286158904071336871570042, 3.91276176140831411905610703583, 4.81364852128904205695527148469, 4.88229729551507725124123543426, 5.17546941442672627763060288287, 5.22757285037087253942056979328, 6.07402294226655892280926950871, 6.25375694774902370799747778372, 6.39344764440031256656743352707, 6.43430435568886477839358906226, 7.31945320421481453145812182693, 7.61053511825945128538483787804, 7.66996318468826532431755722387, 7.69408136750643354346208589279, 8.400597831741162369422740489997