Dirichlet series
| L(s) = 1 | + 405·3-s − 772·5-s + 38·7-s + 9.84e4·9-s + 4.11e4·11-s − 2.24e4·13-s − 3.12e5·15-s − 1.13e3·17-s − 6.64e5·19-s + 1.53e4·21-s − 3.69e5·23-s − 4.17e6·25-s + 1.86e7·27-s − 7.39e6·29-s − 9.14e6·31-s + 1.66e7·33-s − 2.93e4·35-s − 1.19e7·37-s − 9.10e6·39-s − 8.47e6·41-s − 8.94e6·43-s − 7.59e7·45-s + 5.05e6·47-s − 8.10e7·49-s − 4.57e5·51-s + 3.14e7·53-s − 3.17e7·55-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 0.552·5-s + 0.00598·7-s + 5·9-s + 0.846·11-s − 0.218·13-s − 1.59·15-s − 0.00328·17-s − 1.17·19-s + 0.0172·21-s − 0.275·23-s − 2.13·25-s + 6.73·27-s − 1.94·29-s − 1.77·31-s + 2.44·33-s − 0.00330·35-s − 1.04·37-s − 0.630·39-s − 0.468·41-s − 0.399·43-s − 2.76·45-s + 0.151·47-s − 2.00·49-s − 0.00947·51-s + 0.547·53-s − 0.467·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{35} \cdot 3^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{35} \cdot 3^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(2^{35} \cdot 3^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.02582\times 10^{11}\) |
| Root analytic conductor: | \(14.0632\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 2^{35} \cdot 3^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )\) |
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 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{5} \) | |
| good | 5 | $C_2 \wr S_5$ | \( 1 + 772 T + 4769113 T^{2} + 3119327184 T^{3} + 2755862469586 p T^{4} + 381647220137752 p^{2} T^{5} + 2755862469586 p^{10} T^{6} + 3119327184 p^{18} T^{7} + 4769113 p^{27} T^{8} + 772 p^{36} T^{9} + p^{45} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 38 T + 81076683 T^{2} + 9715165848 p T^{3} + 36667571128674 p^{2} T^{4} + 16434539979105156 p^{3} T^{5} + 36667571128674 p^{11} T^{6} + 9715165848 p^{19} T^{7} + 81076683 p^{27} T^{8} - 38 p^{36} T^{9} + p^{45} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 - 41100 T + 6615446391 T^{2} - 264450767569808 T^{3} + 22247372238638525146 T^{4} - \)\(80\!\cdots\!68\)\( T^{5} + 22247372238638525146 p^{9} T^{6} - 264450767569808 p^{18} T^{7} + 6615446391 p^{27} T^{8} - 41100 p^{36} T^{9} + p^{45} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 22486 T + 10397042057 T^{2} - 800227237332184 T^{3} + \)\(16\!\cdots\!86\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!86\)\( p^{9} T^{6} - 800227237332184 p^{18} T^{7} + 10397042057 p^{27} T^{8} + 22486 p^{36} T^{9} + p^{45} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 1130 T + 184610159869 T^{2} + 51018429507312504 T^{3} + \)\(31\!\cdots\!62\)\( T^{4} + \)\(55\!\cdots\!60\)\( T^{5} + \)\(31\!\cdots\!62\)\( p^{9} T^{6} + 51018429507312504 p^{18} T^{7} + 184610159869 p^{27} T^{8} + 1130 p^{36} T^{9} + p^{45} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 664712 T + 577002263231 T^{2} + 368626785895985248 T^{3} + \)\(32\!\cdots\!82\)\( T^{4} + \)\(15\!\cdots\!84\)\( T^{5} + \)\(32\!\cdots\!82\)\( p^{9} T^{6} + 368626785895985248 p^{18} T^{7} + 577002263231 p^{27} T^{8} + 664712 p^{36} T^{9} + p^{45} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 + 369972 T + 7525030157523 T^{2} + 2722898959045623088 T^{3} + \)\(24\!\cdots\!02\)\( T^{4} + \)\(74\!\cdots\!52\)\( T^{5} + \)\(24\!\cdots\!02\)\( p^{9} T^{6} + 2722898959045623088 p^{18} T^{7} + 7525030157523 p^{27} T^{8} + 369972 p^{36} T^{9} + p^{45} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 + 7390736 T + 38239787375089 T^{2} + \)\(16\!\cdots\!36\)\( T^{3} + \)\(67\!\cdots\!38\)\( T^{4} + \)\(27\!\cdots\!76\)\( T^{5} + \)\(67\!\cdots\!38\)\( p^{9} T^{6} + \)\(16\!\cdots\!36\)\( p^{18} T^{7} + 38239787375089 p^{27} T^{8} + 7390736 p^{36} T^{9} + p^{45} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 9149938 T + 64830852324419 T^{2} + 83090231898530573512 T^{3} - \)\(12\!\cdots\!42\)\( T^{4} - \)\(11\!\cdots\!76\)\( T^{5} - \)\(12\!\cdots\!42\)\( p^{9} T^{6} + 83090231898530573512 p^{18} T^{7} + 64830852324419 p^{27} T^{8} + 9149938 p^{36} T^{9} + p^{45} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 11922058 T + 470401367804865 T^{2} + \)\(54\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!72\)\( T^{5} + \)\(10\!\cdots\!38\)\( p^{9} T^{6} + \)\(54\!\cdots\!52\)\( p^{18} T^{7} + 470401367804865 p^{27} T^{8} + 11922058 p^{36} T^{9} + p^{45} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 8471746 T + 925916917222261 T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!34\)\( T^{4} + \)\(42\!\cdots\!08\)\( T^{5} + \)\(48\!\cdots\!34\)\( p^{9} T^{6} + \)\(10\!\cdots\!20\)\( p^{18} T^{7} + 925916917222261 p^{27} T^{8} + 8471746 p^{36} T^{9} + p^{45} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 + 8948896 T + 780215096111 p^{2} T^{2} + \)\(36\!\cdots\!72\)\( T^{3} + \)\(93\!\cdots\!86\)\( T^{4} - \)\(70\!\cdots\!16\)\( T^{5} + \)\(93\!\cdots\!86\)\( p^{9} T^{6} + \)\(36\!\cdots\!72\)\( p^{18} T^{7} + 780215096111 p^{29} T^{8} + 8948896 p^{36} T^{9} + p^{45} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 - 5051660 T + 1959867314605579 T^{2} - \)\(57\!\cdots\!32\)\( T^{3} + \)\(94\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!68\)\( T^{5} + \)\(94\!\cdots\!86\)\( p^{9} T^{6} - \)\(57\!\cdots\!32\)\( p^{18} T^{7} + 1959867314605579 p^{27} T^{8} - 5051660 p^{36} T^{9} + p^{45} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 - 31431984 T + 8965440401334313 T^{2} - \)\(27\!\cdots\!48\)\( T^{3} + \)\(44\!\cdots\!82\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(44\!\cdots\!82\)\( p^{9} T^{6} - \)\(27\!\cdots\!48\)\( p^{18} T^{7} + 8965440401334313 p^{27} T^{8} - 31431984 p^{36} T^{9} + p^{45} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - 204260948 T + 41651072341382375 T^{2} - \)\(48\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!54\)\( T^{4} - \)\(55\!\cdots\!24\)\( T^{5} + \)\(62\!\cdots\!54\)\( p^{9} T^{6} - \)\(48\!\cdots\!20\)\( p^{18} T^{7} + 41651072341382375 p^{27} T^{8} - 204260948 p^{36} T^{9} + p^{45} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 190850874 T + 57359668620683641 T^{2} + \)\(78\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!54\)\( T^{4} + \)\(13\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!54\)\( p^{9} T^{6} + \)\(78\!\cdots\!40\)\( p^{18} T^{7} + 57359668620683641 p^{27} T^{8} + 190850874 p^{36} T^{9} + p^{45} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 - 274483500 T + 106015185970286671 T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + \)\(33\!\cdots\!54\)\( T^{4} - \)\(30\!\cdots\!04\)\( T^{5} + \)\(33\!\cdots\!54\)\( p^{9} T^{6} - \)\(13\!\cdots\!28\)\( p^{18} T^{7} + 106015185970286671 p^{27} T^{8} - 274483500 p^{36} T^{9} + p^{45} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 162722908 T + 54115803211925059 T^{2} - \)\(26\!\cdots\!36\)\( T^{3} - \)\(42\!\cdots\!02\)\( T^{4} - \)\(26\!\cdots\!28\)\( T^{5} - \)\(42\!\cdots\!02\)\( p^{9} T^{6} - \)\(26\!\cdots\!36\)\( p^{18} T^{7} + 54115803211925059 p^{27} T^{8} + 162722908 p^{36} T^{9} + p^{45} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 508927538 T + 324572353550856597 T^{2} + \)\(10\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} + \)\(84\!\cdots\!52\)\( T^{5} + \)\(37\!\cdots\!78\)\( p^{9} T^{6} + \)\(10\!\cdots\!96\)\( p^{18} T^{7} + 324572353550856597 p^{27} T^{8} + 508927538 p^{36} T^{9} + p^{45} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + 491411266 T + 314646363855190835 T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(62\!\cdots\!90\)\( T^{4} + \)\(19\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!90\)\( p^{9} T^{6} + \)\(11\!\cdots\!96\)\( p^{18} T^{7} + 314646363855190835 p^{27} T^{8} + 491411266 p^{36} T^{9} + p^{45} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 - 766279260 T + 707111048975862559 T^{2} - \)\(42\!\cdots\!84\)\( T^{3} + \)\(24\!\cdots\!54\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} + \)\(24\!\cdots\!54\)\( p^{9} T^{6} - \)\(42\!\cdots\!84\)\( p^{18} T^{7} + 707111048975862559 p^{27} T^{8} - 766279260 p^{36} T^{9} + p^{45} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 + 954097990 T + 907217335575083045 T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!64\)\( T^{5} + \)\(28\!\cdots\!70\)\( p^{9} T^{6} + \)\(41\!\cdots\!40\)\( p^{18} T^{7} + 907217335575083045 p^{27} T^{8} + 954097990 p^{36} T^{9} + p^{45} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 677085326 T + 2429584862807454413 T^{2} + \)\(11\!\cdots\!88\)\( T^{3} + \)\(31\!\cdots\!18\)\( T^{4} + \)\(12\!\cdots\!64\)\( T^{5} + \)\(31\!\cdots\!18\)\( p^{9} T^{6} + \)\(11\!\cdots\!88\)\( p^{18} T^{7} + 2429584862807454413 p^{27} T^{8} + 677085326 p^{36} T^{9} + p^{45} T^{10} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.82538011664825526601223905995, −5.59063993945673424554744401675, −5.53252303139876203209538893902, −5.44722452963141859552286068473, −5.27314610666690047500191808659, −4.66964844170006484196477152912, −4.44249137668299446081966450095, −4.44084109700196077189275199361, −4.23681782224890250530069081501, −3.89563283653776124238634081292, −3.74785709722174722190442945594, −3.72617489850809642039729800967, −3.42135561282904940940132220332, −3.39198146402209521981493590440, −3.15292350524527534683077638777, −2.60589625894087866184235501316, −2.54657408127349358583183957124, −2.36251968585045367604155709884, −2.16128726268812492902260032210, −2.07882918772510583510346419318, −1.53823095096431889059739448067, −1.49168739596438614322766255581, −1.47271062461248909238258883567, −1.10794941666152551232830507883, −1.06954436895772972613862566870, 0, 0, 0, 0, 0, 1.06954436895772972613862566870, 1.10794941666152551232830507883, 1.47271062461248909238258883567, 1.49168739596438614322766255581, 1.53823095096431889059739448067, 2.07882918772510583510346419318, 2.16128726268812492902260032210, 2.36251968585045367604155709884, 2.54657408127349358583183957124, 2.60589625894087866184235501316, 3.15292350524527534683077638777, 3.39198146402209521981493590440, 3.42135561282904940940132220332, 3.72617489850809642039729800967, 3.74785709722174722190442945594, 3.89563283653776124238634081292, 4.23681782224890250530069081501, 4.44084109700196077189275199361, 4.44249137668299446081966450095, 4.66964844170006484196477152912, 5.27314610666690047500191808659, 5.44722452963141859552286068473, 5.53252303139876203209538893902, 5.59063993945673424554744401675, 5.82538011664825526601223905995