Dirichlet series
| L(s) = 1 | − 512·2-s + 8.74e3·3-s + 1.63e5·4-s − 2.65e3·5-s − 4.47e6·6-s + 2.25e6·7-s − 4.19e7·8-s + 4.78e7·9-s + 1.35e6·10-s − 4.15e7·11-s + 1.43e9·12-s + 2.50e8·13-s − 1.15e9·14-s − 2.31e7·15-s + 9.39e9·16-s + 4.72e8·17-s − 2.44e10·18-s − 8.76e9·19-s − 4.34e8·20-s + 1.97e10·21-s + 2.12e10·22-s + 1.45e10·23-s − 3.66e11·24-s − 5.80e10·25-s − 1.28e11·26-s + 2.09e11·27-s + 3.70e11·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 0.0151·5-s − 6.53·6-s + 1.03·7-s − 7.07·8-s + 10/3·9-s + 0.0429·10-s − 0.643·11-s + 11.5·12-s + 1.10·13-s − 2.93·14-s − 0.0350·15-s + 35/4·16-s + 0.279·17-s − 9.42·18-s − 2.25·19-s − 0.0758·20-s + 2.39·21-s + 1.82·22-s + 0.894·23-s − 16.3·24-s − 1.90·25-s − 3.13·26-s + 3.84·27-s + 5.18·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.53460\times 10^{8}\) |
| Root analytic conductor: | \(10.5499\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 37015056,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(6.392895388\) |
| \(L(\frac12)\) | \(\approx\) | \(6.392895388\) |
| \(L(\frac{17}{2})\) | 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 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 106 p^{2} T + 11615222544 p T^{2} - 281217937789122 p^{2} T^{3} + 2394166825613473198 p^{4} T^{4} - 281217937789122 p^{17} T^{5} + 11615222544 p^{31} T^{6} + 106 p^{47} T^{7} + p^{60} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2259438 T + 958069795460 p T^{2} - 382721843654377158 p^{2} T^{3} + \)\(16\!\cdots\!10\)\( p^{3} T^{4} - 382721843654377158 p^{17} T^{5} + 958069795460 p^{31} T^{6} - 2259438 p^{45} T^{7} + p^{60} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 41589496 T + 292519087702164 p T^{2} + \)\(17\!\cdots\!48\)\( p^{2} T^{3} + \)\(28\!\cdots\!26\)\( p^{3} T^{4} + \)\(17\!\cdots\!48\)\( p^{17} T^{5} + 292519087702164 p^{31} T^{6} + 41589496 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 27800608 p T + 8173694682357583212 T^{2} - \)\(36\!\cdots\!64\)\( T^{3} + \)\(32\!\cdots\!34\)\( T^{4} - \)\(36\!\cdots\!64\)\( p^{15} T^{5} + 8173694682357583212 p^{30} T^{6} - 27800608 p^{46} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 8769424494 T + 59455812990694654916 T^{2} + \)\(24\!\cdots\!38\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!38\)\( p^{15} T^{5} + 59455812990694654916 p^{30} T^{6} + 8769424494 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 14598610608 T + \)\(61\!\cdots\!68\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(20\!\cdots\!54\)\( T^{4} - \)\(11\!\cdots\!28\)\( p^{15} T^{5} + \)\(61\!\cdots\!68\)\( p^{30} T^{6} - 14598610608 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 57157147020 T + \)\(21\!\cdots\!12\)\( T^{2} - \)\(93\!\cdots\!36\)\( T^{3} + \)\(23\!\cdots\!46\)\( T^{4} - \)\(93\!\cdots\!36\)\( p^{15} T^{5} + \)\(21\!\cdots\!12\)\( p^{30} T^{6} - 57157147020 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 50348256398 T + \)\(59\!\cdots\!72\)\( T^{2} - \)\(12\!\cdots\!90\)\( T^{3} + \)\(16\!\cdots\!42\)\( T^{4} - \)\(12\!\cdots\!90\)\( p^{15} T^{5} + \)\(59\!\cdots\!72\)\( p^{30} T^{6} - 50348256398 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 546333591492 T + \)\(33\!\cdots\!52\)\( T^{2} + \)\(31\!\cdots\!16\)\( T^{3} + \)\(23\!\cdots\!58\)\( T^{4} + \)\(31\!\cdots\!16\)\( p^{15} T^{5} + \)\(33\!\cdots\!52\)\( p^{30} T^{6} + 546333591492 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 2795029286222 T + \)\(70\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!14\)\( T^{3} + \)\(15\!\cdots\!90\)\( T^{4} - \)\(10\!\cdots\!14\)\( p^{15} T^{5} + \)\(70\!\cdots\!68\)\( p^{30} T^{6} - 2795029286222 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 68181646920 p T + \)\(90\!\cdots\!12\)\( T^{2} - \)\(19\!\cdots\!28\)\( T^{3} + \)\(43\!\cdots\!82\)\( T^{4} - \)\(19\!\cdots\!28\)\( p^{15} T^{5} + \)\(90\!\cdots\!12\)\( p^{30} T^{6} - 68181646920 p^{46} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 749893381796 T + \)\(19\!\cdots\!64\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(29\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!92\)\( p^{15} T^{5} + \)\(19\!\cdots\!64\)\( p^{30} T^{6} - 749893381796 p^{45} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 18224949214424 T + \)\(23\!\cdots\!60\)\( T^{2} - \)\(25\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!82\)\( T^{4} - \)\(25\!\cdots\!16\)\( p^{15} T^{5} + \)\(23\!\cdots\!60\)\( p^{30} T^{6} - 18224949214424 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 38638125608492 T + \)\(16\!\cdots\!56\)\( T^{2} + \)\(38\!\cdots\!32\)\( T^{3} + \)\(90\!\cdots\!50\)\( T^{4} + \)\(38\!\cdots\!32\)\( p^{15} T^{5} + \)\(16\!\cdots\!56\)\( p^{30} T^{6} + 38638125608492 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 24564225312276 T + \)\(17\!\cdots\!96\)\( T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(32\!\cdots\!56\)\( p^{15} T^{5} + \)\(17\!\cdots\!96\)\( p^{30} T^{6} + 24564225312276 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 94206416835706 T + \)\(11\!\cdots\!52\)\( T^{2} + \)\(68\!\cdots\!74\)\( T^{3} + \)\(43\!\cdots\!74\)\( T^{4} + \)\(68\!\cdots\!74\)\( p^{15} T^{5} + \)\(11\!\cdots\!52\)\( p^{30} T^{6} + 94206416835706 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 126781073140592 T + \)\(22\!\cdots\!48\)\( T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(19\!\cdots\!90\)\( T^{4} + \)\(21\!\cdots\!64\)\( p^{15} T^{5} + \)\(22\!\cdots\!48\)\( p^{30} T^{6} + 126781073140592 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 118248778504108 T + \)\(34\!\cdots\!16\)\( T^{2} + \)\(29\!\cdots\!08\)\( T^{3} + \)\(46\!\cdots\!14\)\( T^{4} + \)\(29\!\cdots\!08\)\( p^{15} T^{5} + \)\(34\!\cdots\!16\)\( p^{30} T^{6} + 118248778504108 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 195093212840776 T + \)\(94\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{3} + \)\(35\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!72\)\( p^{15} T^{5} + \)\(94\!\cdots\!08\)\( p^{30} T^{6} + 195093212840776 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 160896157851060 T + \)\(14\!\cdots\!84\)\( T^{2} - \)\(26\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!90\)\( T^{4} - \)\(26\!\cdots\!44\)\( p^{15} T^{5} + \)\(14\!\cdots\!84\)\( p^{30} T^{6} - 160896157851060 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 641924370809666 T + \)\(80\!\cdots\!12\)\( T^{2} - \)\(33\!\cdots\!18\)\( T^{3} + \)\(21\!\cdots\!50\)\( T^{4} - \)\(33\!\cdots\!18\)\( p^{15} T^{5} + \)\(80\!\cdots\!12\)\( p^{30} T^{6} - 641924370809666 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 518459139629100 T + \)\(15\!\cdots\!76\)\( T^{2} + \)\(55\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} + \)\(55\!\cdots\!92\)\( p^{15} T^{5} + \)\(15\!\cdots\!76\)\( p^{30} T^{6} + 518459139629100 p^{45} T^{7} + p^{60} T^{8} \) | |
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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
−8.112082526021315416958744135285, −7.45481772640708695557729143624, −7.29673063336400998127174354487, −7.27809166981600855105575615015, −7.11693408151987607186314192782, −6.24982327086610617532149467964, −6.07087065950439568171592337268, −5.94448331355922535789192021286, −5.73718310512166740941399783233, −4.83704881388207201099538017342, −4.40755246293719338976764523060, −4.34209368913292167787967056359, −4.09370315324805953848665624005, −3.34425473257619744268056289095, −3.14594296721613402510924488829, −3.03728420884016837095589479406, −2.65257849301782188177707747667, −2.08650782933667812388681193218, −1.95740437913456701528897329106, −1.89949641567519350385394389470, −1.70659730325107291447548401275, −1.05399338350749981708695973476, −0.936567445934281370613593546867, −0.50178702983579439891250339136, −0.37845357167173186939735891819, 0.37845357167173186939735891819, 0.50178702983579439891250339136, 0.936567445934281370613593546867, 1.05399338350749981708695973476, 1.70659730325107291447548401275, 1.89949641567519350385394389470, 1.95740437913456701528897329106, 2.08650782933667812388681193218, 2.65257849301782188177707747667, 3.03728420884016837095589479406, 3.14594296721613402510924488829, 3.34425473257619744268056289095, 4.09370315324805953848665624005, 4.34209368913292167787967056359, 4.40755246293719338976764523060, 4.83704881388207201099538017342, 5.73718310512166740941399783233, 5.94448331355922535789192021286, 6.07087065950439568171592337268, 6.24982327086610617532149467964, 7.11693408151987607186314192782, 7.27809166981600855105575615015, 7.29673063336400998127174354487, 7.45481772640708695557729143624, 8.112082526021315416958744135285