Dirichlet series
| L(s) = 1 | − 19·2-s + 18·3-s + 21·4-s + 2.50e3·5-s − 342·6-s − 5.03e3·8-s − 3.65e4·9-s − 4.75e4·10-s + 8.24e4·11-s + 378·12-s + 7.29e4·13-s + 4.50e4·15-s − 8.43e4·16-s + 3.57e5·17-s + 6.93e5·18-s − 3.00e5·19-s + 5.25e4·20-s − 1.56e6·22-s − 2.34e6·23-s − 9.06e4·24-s + 3.90e6·25-s − 1.38e6·26-s + 6.36e5·27-s − 1.37e6·29-s − 8.55e5·30-s + 4.59e6·31-s + 4.69e6·32-s + ⋯ |
| L(s) = 1 | − 0.839·2-s + 0.128·3-s + 0.0410·4-s + 1.78·5-s − 0.107·6-s − 0.434·8-s − 1.85·9-s − 1.50·10-s + 1.69·11-s + 0.00526·12-s + 0.708·13-s + 0.229·15-s − 0.321·16-s + 1.03·17-s + 1.55·18-s − 0.529·19-s + 0.0733·20-s − 1.42·22-s − 1.74·23-s − 0.0557·24-s + 2·25-s − 0.594·26-s + 0.230·27-s − 0.360·29-s − 0.192·30-s + 0.892·31-s + 0.790·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(5^{4} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.53521\times 10^{8}\) |
| Root analytic conductor: | \(11.2331\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 5^{4} \cdot 7^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 5 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 7 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 + 19 T + 85 p^{2} T^{2} + 1387 p^{3} T^{3} + 11991 p^{5} T^{4} + 1387 p^{12} T^{5} + 85 p^{20} T^{6} + 19 p^{27} T^{7} + p^{36} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 2 p^{2} T + 4093 p^{2} T^{2} - 24154 p^{4} T^{3} + 11605708 p^{4} T^{4} - 24154 p^{13} T^{5} + 4093 p^{20} T^{6} - 2 p^{29} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 82438 T + 8266219573 T^{2} - 478792229404886 T^{3} + 28732371262031407500 T^{4} - 478792229404886 p^{9} T^{5} + 8266219573 p^{18} T^{6} - 82438 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 72962 T + 28727078025 T^{2} - 1965410763776002 T^{3} + \)\(42\!\cdots\!72\)\( T^{4} - 1965410763776002 p^{9} T^{5} + 28727078025 p^{18} T^{6} - 72962 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 357542 T + 468292069057 T^{2} - 118264476533781310 T^{3} + \)\(82\!\cdots\!36\)\( T^{4} - 118264476533781310 p^{9} T^{5} + 468292069057 p^{18} T^{6} - 357542 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 15828 p T + 621418286736 T^{2} + 177926116548096924 T^{3} + \)\(29\!\cdots\!06\)\( T^{4} + 177926116548096924 p^{9} T^{5} + 621418286736 p^{18} T^{6} + 15828 p^{28} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 2340836 T + 7030586174368 T^{2} + 11439800875282843060 T^{3} + \)\(19\!\cdots\!46\)\( T^{4} + 11439800875282843060 p^{9} T^{5} + 7030586174368 p^{18} T^{6} + 2340836 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1372022 T + 46742390138801 T^{2} + 1810211425259735326 p T^{3} + \)\(96\!\cdots\!16\)\( T^{4} + 1810211425259735326 p^{10} T^{5} + 46742390138801 p^{18} T^{6} + 1372022 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 4590888 T + 56000638943212 T^{2} - \)\(20\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!50\)\( T^{4} - \)\(20\!\cdots\!64\)\( p^{9} T^{5} + 56000638943212 p^{18} T^{6} - 4590888 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 38868456 T + 975525049444828 T^{2} + \)\(17\!\cdots\!52\)\( T^{3} + \)\(22\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!52\)\( p^{9} T^{5} + 975525049444828 p^{18} T^{6} + 38868456 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 57287084 T + 1879166479908184 T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(84\!\cdots\!30\)\( T^{4} + \)\(43\!\cdots\!36\)\( p^{9} T^{5} + 1879166479908184 p^{18} T^{6} + 57287084 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 43403452 T + 1083079781009760 T^{2} + \)\(21\!\cdots\!52\)\( T^{3} + \)\(54\!\cdots\!02\)\( T^{4} + \)\(21\!\cdots\!52\)\( p^{9} T^{5} + 1083079781009760 p^{18} T^{6} + 43403452 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 91822222 T + 5137103108541341 T^{2} + \)\(19\!\cdots\!82\)\( T^{3} + \)\(68\!\cdots\!04\)\( T^{4} + \)\(19\!\cdots\!82\)\( p^{9} T^{5} + 5137103108541341 p^{18} T^{6} + 91822222 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 61086884 T + 9515369140509784 T^{2} + \)\(52\!\cdots\!16\)\( T^{3} + \)\(40\!\cdots\!54\)\( T^{4} + \)\(52\!\cdots\!16\)\( p^{9} T^{5} + 9515369140509784 p^{18} T^{6} + 61086884 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 72569680 T + 15506079620308076 T^{2} + \)\(53\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} + \)\(53\!\cdots\!60\)\( p^{9} T^{5} + 15506079620308076 p^{18} T^{6} + 72569680 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 225681036 T + 40564239782954584 T^{2} - \)\(44\!\cdots\!64\)\( T^{3} + \)\(52\!\cdots\!70\)\( T^{4} - \)\(44\!\cdots\!64\)\( p^{9} T^{5} + 40564239782954584 p^{18} T^{6} - 225681036 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 4429720 T + 72492995180634444 T^{2} + \)\(22\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!30\)\( T^{4} + \)\(22\!\cdots\!64\)\( p^{9} T^{5} + 72492995180634444 p^{18} T^{6} + 4429720 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 470468984 T + 190170620017254236 T^{2} + \)\(49\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!70\)\( T^{4} + \)\(49\!\cdots\!64\)\( p^{9} T^{5} + 190170620017254236 p^{18} T^{6} + 470468984 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 168326464 T + 163795153067496204 T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(18\!\cdots\!36\)\( p^{9} T^{5} + 163795153067496204 p^{18} T^{6} - 168326464 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 598805646 T + 409167627137545941 T^{2} + \)\(15\!\cdots\!22\)\( T^{3} + \)\(69\!\cdots\!36\)\( T^{4} + \)\(15\!\cdots\!22\)\( p^{9} T^{5} + 409167627137545941 p^{18} T^{6} + 598805646 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 1159074304 T + 958573764116851484 T^{2} - \)\(57\!\cdots\!36\)\( T^{3} + \)\(28\!\cdots\!34\)\( T^{4} - \)\(57\!\cdots\!36\)\( p^{9} T^{5} + 958573764116851484 p^{18} T^{6} - 1159074304 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1380153012 T + 1820467990344974776 T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(99\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!84\)\( p^{9} T^{5} + 1820467990344974776 p^{18} T^{6} - 1380153012 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 754874082 T + 1211193206499632841 T^{2} + \)\(57\!\cdots\!42\)\( T^{3} + \)\(91\!\cdots\!84\)\( T^{4} + \)\(57\!\cdots\!42\)\( p^{9} T^{5} + 1211193206499632841 p^{18} T^{6} + 754874082 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.87527268789314673537870257093, −7.52290828035803538041778604698, −6.89072641173905961171068901321, −6.78120802965870354910673142680, −6.50215523600689673735781414915, −6.40295853477738356428453519418, −6.28391084161706052994174747267, −6.06236459484387687899184614289, −5.45876699553476753195679150815, −5.36413011145950256525440617842, −5.16605915513500069722533077138, −4.94608169664348340403796018669, −4.62246520251672496135209777647, −4.05279501209460302322339699041, −3.58337847606243240046132231333, −3.50822207619488338138630397791, −3.40541970845069268061049248723, −2.92851339096159591547879242397, −2.74906069776721413725894650417, −2.16796398414200331493365067186, −1.88904320934142330812663533298, −1.81360021828413652287151393157, −1.50630626633573607450862130308, −1.09214550887724734976111721919, −1.06467517233879691423110603447, 0, 0, 0, 0, 1.06467517233879691423110603447, 1.09214550887724734976111721919, 1.50630626633573607450862130308, 1.81360021828413652287151393157, 1.88904320934142330812663533298, 2.16796398414200331493365067186, 2.74906069776721413725894650417, 2.92851339096159591547879242397, 3.40541970845069268061049248723, 3.50822207619488338138630397791, 3.58337847606243240046132231333, 4.05279501209460302322339699041, 4.62246520251672496135209777647, 4.94608169664348340403796018669, 5.16605915513500069722533077138, 5.36413011145950256525440617842, 5.45876699553476753195679150815, 6.06236459484387687899184614289, 6.28391084161706052994174747267, 6.40295853477738356428453519418, 6.50215523600689673735781414915, 6.78120802965870354910673142680, 6.89072641173905961171068901321, 7.52290828035803538041778604698, 7.87527268789314673537870257093