Dirichlet series
| L(s) = 1 | + 64·2-s + 2.56e3·4-s − 1.64e3·5-s − 9.60e3·7-s + 8.19e4·8-s − 1.05e5·10-s + 1.05e4·11-s − 1.08e5·13-s − 6.14e5·14-s + 2.29e6·16-s + 5.29e5·17-s + 4.67e5·19-s − 4.22e6·20-s + 6.74e5·22-s − 1.18e6·23-s − 1.91e6·25-s − 6.94e6·26-s − 2.45e7·28-s − 1.94e6·29-s + 1.11e7·31-s + 5.87e7·32-s + 3.38e7·34-s + 1.58e7·35-s + 6.22e6·37-s + 2.99e7·38-s − 1.35e8·40-s − 3.55e7·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 1.17·5-s − 1.51·7-s + 7.07·8-s − 3.33·10-s + 0.217·11-s − 1.05·13-s − 4.27·14-s + 35/4·16-s + 1.53·17-s + 0.822·19-s − 5.89·20-s + 0.614·22-s − 0.883·23-s − 0.979·25-s − 2.98·26-s − 7.55·28-s − 0.510·29-s + 2.16·31-s + 9.89·32-s + 4.34·34-s + 1.78·35-s + 0.545·37-s + 2.32·38-s − 8.34·40-s − 1.96·41-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: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 + 1649 T + 4632059 T^{2} + 7198504766 T^{3} + 494552206468 p^{2} T^{4} + 7198504766 p^{9} T^{5} + 4632059 p^{18} T^{6} + 1649 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 10544 T + 4501818704 T^{2} + 48571520009248 T^{3} + 10232662479928715758 T^{4} + 48571520009248 p^{9} T^{5} + 4501818704 p^{18} T^{6} - 10544 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 108585 T + 28739498746 T^{2} + 237050146972971 p T^{3} + \)\(40\!\cdots\!10\)\( T^{4} + 237050146972971 p^{10} T^{5} + 28739498746 p^{18} T^{6} + 108585 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 529388 T + 404925326390 T^{2} - 153933571996748792 T^{3} + \)\(65\!\cdots\!67\)\( T^{4} - 153933571996748792 p^{9} T^{5} + 404925326390 p^{18} T^{6} - 529388 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 467258 T + 669417733576 T^{2} - 386087640070361474 T^{3} + \)\(14\!\cdots\!34\)\( p T^{4} - 386087640070361474 p^{9} T^{5} + 669417733576 p^{18} T^{6} - 467258 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 1185163 T + 818553032606 T^{2} + 2935199747877112603 T^{3} + \)\(59\!\cdots\!90\)\( T^{4} + 2935199747877112603 p^{9} T^{5} + 818553032606 p^{18} T^{6} + 1185163 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1945301 T + 19034150961440 T^{2} + \)\(11\!\cdots\!55\)\( T^{3} + \)\(22\!\cdots\!22\)\( T^{4} + \)\(11\!\cdots\!55\)\( p^{9} T^{5} + 19034150961440 p^{18} T^{6} + 1945301 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 11130777 T + 142428040331926 T^{2} - \)\(91\!\cdots\!29\)\( T^{3} + \)\(61\!\cdots\!62\)\( T^{4} - \)\(91\!\cdots\!29\)\( p^{9} T^{5} + 142428040331926 p^{18} T^{6} - 11130777 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 6222809 T + 394596874081837 T^{2} - \)\(24\!\cdots\!34\)\( T^{3} + \)\(70\!\cdots\!74\)\( T^{4} - \)\(24\!\cdots\!34\)\( p^{9} T^{5} + 394596874081837 p^{18} T^{6} - 6222809 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 35545545 T + 1271356128579209 T^{2} + \)\(28\!\cdots\!26\)\( T^{3} + \)\(63\!\cdots\!06\)\( T^{4} + \)\(28\!\cdots\!26\)\( p^{9} T^{5} + 1271356128579209 p^{18} T^{6} + 35545545 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 7202206 T + 704053605469636 T^{2} - \)\(47\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} - \)\(47\!\cdots\!60\)\( p^{9} T^{5} + 704053605469636 p^{18} T^{6} + 7202206 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 20761503 T + 3264933800921081 T^{2} - \)\(59\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!26\)\( T^{4} - \)\(59\!\cdots\!40\)\( p^{9} T^{5} + 3264933800921081 p^{18} T^{6} - 20761503 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 98635821 T + 3374566731728870 T^{2} + \)\(19\!\cdots\!99\)\( T^{3} + \)\(18\!\cdots\!42\)\( T^{4} + \)\(19\!\cdots\!99\)\( p^{9} T^{5} + 3374566731728870 p^{18} T^{6} + 98635821 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 33235282 T + 19102467191049488 T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!17\)\( T^{4} - \)\(10\!\cdots\!88\)\( p^{9} T^{5} + 19102467191049488 p^{18} T^{6} - 33235282 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 105743430 T + 23431283170423384 T^{2} - \)\(26\!\cdots\!82\)\( T^{3} + \)\(28\!\cdots\!66\)\( T^{4} - \)\(26\!\cdots\!82\)\( p^{9} T^{5} + 23431283170423384 p^{18} T^{6} - 105743430 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 155997695 T + 61050802496304124 T^{2} + \)\(13\!\cdots\!79\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} + \)\(13\!\cdots\!79\)\( p^{9} T^{5} + 61050802496304124 p^{18} T^{6} + 155997695 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 154553551 T + 87446999330176088 T^{2} + \)\(14\!\cdots\!55\)\( T^{3} + \)\(55\!\cdots\!74\)\( T^{4} + \)\(14\!\cdots\!55\)\( p^{9} T^{5} + 87446999330176088 p^{18} T^{6} + 154553551 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 696521780 T + 401518814316275764 T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!26\)\( T^{4} + \)\(13\!\cdots\!40\)\( p^{9} T^{5} + 401518814316275764 p^{18} T^{6} + 696521780 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 477454197 T + 498948341872602181 T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + \)\(88\!\cdots\!46\)\( T^{4} + \)\(15\!\cdots\!28\)\( p^{9} T^{5} + 498948341872602181 p^{18} T^{6} + 477454197 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 835952377 T + 941011679554486571 T^{2} + \)\(47\!\cdots\!32\)\( T^{3} + \)\(28\!\cdots\!12\)\( T^{4} + \)\(47\!\cdots\!32\)\( p^{9} T^{5} + 941011679554486571 p^{18} T^{6} + 835952377 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 614491047 T + 505724364858976466 T^{2} + \)\(92\!\cdots\!83\)\( T^{3} - \)\(25\!\cdots\!10\)\( T^{4} + \)\(92\!\cdots\!83\)\( p^{9} T^{5} + 505724364858976466 p^{18} T^{6} - 614491047 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1381829600 T + 3133790128311996664 T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!98\)\( T^{4} + \)\(31\!\cdots\!60\)\( p^{9} T^{5} + 3133790128311996664 p^{18} T^{6} + 1381829600 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.25038119660707312199755682142, −6.64057041898223226025871368975, −6.58238125589017940969838181427, −6.39875071270651042377805217147, −6.17771042090710638917432010264, −5.80292576747644359294199387328, −5.55358034351862063162510024533, −5.36219503554539361796361069072, −5.34355548452079656323435190717, −4.88612026756452769935368495528, −4.41908623571433917873812857556, −4.41471716040724733945798462010, −4.20310342118188303267183953016, −3.80238913213783229590061072550, −3.60216445811039499872390300228, −3.54002630725946186847844426639, −3.24495566300452585947881095342, −2.80563175369842746740634855248, −2.58777044410763837481144184178, −2.55345021057503973769792041339, −2.43467750099797987441832159918, −1.45370504712169782023741481078, −1.44602375891023670686849823407, −1.27457648956090180640534866213, −1.09811636223321355305333290257, 0, 0, 0, 0, 1.09811636223321355305333290257, 1.27457648956090180640534866213, 1.44602375891023670686849823407, 1.45370504712169782023741481078, 2.43467750099797987441832159918, 2.55345021057503973769792041339, 2.58777044410763837481144184178, 2.80563175369842746740634855248, 3.24495566300452585947881095342, 3.54002630725946186847844426639, 3.60216445811039499872390300228, 3.80238913213783229590061072550, 4.20310342118188303267183953016, 4.41471716040724733945798462010, 4.41908623571433917873812857556, 4.88612026756452769935368495528, 5.34355548452079656323435190717, 5.36219503554539361796361069072, 5.55358034351862063162510024533, 5.80292576747644359294199387328, 6.17771042090710638917432010264, 6.39875071270651042377805217147, 6.58238125589017940969838181427, 6.64057041898223226025871368975, 7.25038119660707312199755682142