Dirichlet series
| L(s) = 1 | − 64·2-s + 2.56e3·4-s + 2.20e3·5-s + 9.60e3·7-s − 8.19e4·8-s − 1.41e5·10-s + 1.47e4·11-s + 2.14e4·13-s − 6.14e5·14-s + 2.29e6·16-s + 4.85e5·17-s − 4.68e4·19-s + 5.65e6·20-s − 9.46e5·22-s − 4.30e5·23-s − 2.54e6·25-s − 1.37e6·26-s + 2.45e7·28-s + 2.58e6·29-s − 6.22e6·31-s − 5.87e7·32-s − 3.10e7·34-s + 2.12e7·35-s + 8.99e6·37-s + 2.99e6·38-s − 1.80e8·40-s + 2.20e7·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s + 1.57·5-s + 1.51·7-s − 7.07·8-s − 4.46·10-s + 0.304·11-s + 0.208·13-s − 4.27·14-s + 35/4·16-s + 1.41·17-s − 0.0824·19-s + 7.89·20-s − 0.861·22-s − 0.321·23-s − 1.30·25-s − 0.589·26-s + 7.55·28-s + 0.677·29-s − 1.21·31-s − 9.89·32-s − 3.98·34-s + 2.38·35-s + 0.789·37-s + 0.233·38-s − 11.1·40-s + 1.21·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{12} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.43653\times 10^{9}\) |
| Root analytic conductor: | \(13.9529\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(6.171366284\) |
| \(L(\frac12)\) | \(\approx\) | \(6.171366284\) |
| \(L(\frac{11}{2})\) | 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 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2208 T + 7415726 T^{2} - 2406860928 p T^{3} + 860617141179 p^{2} T^{4} - 2406860928 p^{10} T^{5} + 7415726 p^{18} T^{6} - 2208 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 1344 p T + 5325285398 T^{2} - 67924990883520 T^{3} + 17889381886564963563 T^{4} - 67924990883520 p^{9} T^{5} + 5325285398 p^{18} T^{6} - 1344 p^{28} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 21464 T + 21751402636 T^{2} + 339823680844984 T^{3} + 18037851891023757262 p T^{4} + 339823680844984 p^{9} T^{5} + 21751402636 p^{18} T^{6} - 21464 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 485568 T + 213805349348 T^{2} - 90962990510119488 T^{3} + \)\(43\!\cdots\!26\)\( T^{4} - 90962990510119488 p^{9} T^{5} + 213805349348 p^{18} T^{6} - 485568 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 46828 T + 387090774370 T^{2} - 211853405521509608 T^{3} + \)\(91\!\cdots\!35\)\( T^{4} - 211853405521509608 p^{9} T^{5} + 387090774370 p^{18} T^{6} + 46828 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 430944 T + 1108884312086 T^{2} + 1716638356321396416 T^{3} + \)\(57\!\cdots\!51\)\( T^{4} + 1716638356321396416 p^{9} T^{5} + 1108884312086 p^{18} T^{6} + 430944 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 2580000 T + 33906695764316 T^{2} - 44547924253273908192 T^{3} + \)\(58\!\cdots\!94\)\( T^{4} - 44547924253273908192 p^{9} T^{5} + 33906695764316 p^{18} T^{6} - 2580000 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 6228604 T + 62868507171202 T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!79\)\( T^{4} + \)\(20\!\cdots\!20\)\( p^{9} T^{5} + 62868507171202 p^{18} T^{6} + 6228604 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 8995220 T + 497286327190186 T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!55\)\( T^{4} - \)\(34\!\cdots\!20\)\( p^{9} T^{5} + 497286327190186 p^{18} T^{6} - 8995220 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 22028736 T + 1135835206894238 T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!67\)\( p T^{4} - \)\(17\!\cdots\!76\)\( p^{9} T^{5} + 1135835206894238 p^{18} T^{6} - 22028736 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 21954176 T + 1056833312924332 T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(53\!\cdots\!30\)\( T^{4} - \)\(23\!\cdots\!72\)\( p^{9} T^{5} + 1056833312924332 p^{18} T^{6} - 21954176 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 33437472 T + 3207667521708860 T^{2} - \)\(73\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!78\)\( T^{4} - \)\(73\!\cdots\!40\)\( p^{9} T^{5} + 3207667521708860 p^{18} T^{6} - 33437472 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 111929472 T + 12127832562737876 T^{2} - \)\(69\!\cdots\!84\)\( T^{3} + \)\(48\!\cdots\!46\)\( T^{4} - \)\(69\!\cdots\!84\)\( p^{9} T^{5} + 12127832562737876 p^{18} T^{6} - 111929472 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 285328416 T + 55233276708774380 T^{2} - \)\(74\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!66\)\( T^{4} - \)\(74\!\cdots\!40\)\( p^{9} T^{5} + 55233276708774380 p^{18} T^{6} - 285328416 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 36406592 T + 35662036806283636 T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(57\!\cdots\!02\)\( T^{4} - \)\(12\!\cdots\!52\)\( p^{9} T^{5} + 35662036806283636 p^{18} T^{6} - 36406592 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 40399496 T + 54060073617597220 T^{2} - \)\(56\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!10\)\( T^{4} - \)\(56\!\cdots\!60\)\( p^{9} T^{5} + 54060073617597220 p^{18} T^{6} - 40399496 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 99955872 T + 107604681472591430 T^{2} - \)\(31\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!47\)\( T^{4} - \)\(31\!\cdots\!92\)\( p^{9} T^{5} + 107604681472591430 p^{18} T^{6} - 99955872 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 10196576 T + 155819188909561636 T^{2} - \)\(84\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!62\)\( T^{4} - \)\(84\!\cdots\!96\)\( p^{9} T^{5} + 155819188909561636 p^{18} T^{6} - 10196576 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 54438704 T + 3397921806717220 p T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!00\)\( p^{9} T^{5} + 3397921806717220 p^{19} T^{6} - 54438704 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 54880032 T + 663591499736412596 T^{2} + \)\(23\!\cdots\!48\)\( T^{3} + \)\(17\!\cdots\!58\)\( T^{4} + \)\(23\!\cdots\!48\)\( p^{9} T^{5} + 663591499736412596 p^{18} T^{6} + 54880032 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 100706112 T + 1044224680704356894 T^{2} - \)\(38\!\cdots\!48\)\( T^{3} + \)\(48\!\cdots\!07\)\( T^{4} - \)\(38\!\cdots\!48\)\( p^{9} T^{5} + 1044224680704356894 p^{18} T^{6} - 100706112 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1861020208 T + 2144895054783101668 T^{2} + \)\(22\!\cdots\!24\)\( T^{3} + \)\(23\!\cdots\!50\)\( T^{4} + \)\(22\!\cdots\!24\)\( p^{9} T^{5} + 2144895054783101668 p^{18} T^{6} + 1861020208 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.09267615403775442159657742151, −6.38109257329905492506605258202, −6.14074029342938808774787438707, −6.09607073214786177360247759137, −5.93590620721805257802709178572, −5.54385860357265799218467516770, −5.24824854716232862012565750586, −5.19615458504430154961315369682, −5.03249790924599239944672999868, −4.03086772387114794429858649236, −3.95366079974498034009987698067, −3.86409212795905747928376930454, −3.85268169601293083908581701409, −2.77543475087429844351808103709, −2.76852199884891643933101630363, −2.59121082803578481711389434745, −2.26225930093158874459307583476, −2.08843925089611269463505315078, −1.57197541302492573222819572168, −1.53912649805720915814903960655, −1.48488184373283148379254855585, −1.00152612528401486988682499672, −0.63717642148009468277981096171, −0.61572618864076924658009186437, −0.36166734816765442701338749429, 0.36166734816765442701338749429, 0.61572618864076924658009186437, 0.63717642148009468277981096171, 1.00152612528401486988682499672, 1.48488184373283148379254855585, 1.53912649805720915814903960655, 1.57197541302492573222819572168, 2.08843925089611269463505315078, 2.26225930093158874459307583476, 2.59121082803578481711389434745, 2.76852199884891643933101630363, 2.77543475087429844351808103709, 3.85268169601293083908581701409, 3.86409212795905747928376930454, 3.95366079974498034009987698067, 4.03086772387114794429858649236, 5.03249790924599239944672999868, 5.19615458504430154961315369682, 5.24824854716232862012565750586, 5.54385860357265799218467516770, 5.93590620721805257802709178572, 6.09607073214786177360247759137, 6.14074029342938808774787438707, 6.38109257329905492506605258202, 7.09267615403775442159657742151