Dirichlet series
| L(s) = 1 | − 2.04e4·2-s − 7.23e5·3-s + 2.51e8·4-s + 1.48e10·6-s − 4.92e10·7-s − 2.40e12·8-s − 1.36e12·9-s − 8.83e12·11-s − 1.82e14·12-s − 6.76e13·13-s + 1.00e15·14-s + 1.97e16·16-s − 1.54e15·17-s + 2.80e16·18-s + 6.82e15·19-s + 3.56e16·21-s + 1.80e17·22-s − 1.75e17·23-s + 1.74e18·24-s + 1.38e18·26-s + 9.54e17·27-s − 1.23e19·28-s + 2.61e18·29-s + 1.38e18·31-s − 1.45e20·32-s + 6.39e18·33-s + 3.16e19·34-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 0.786·3-s + 15/2·4-s + 2.78·6-s − 1.34·7-s − 12.3·8-s − 1.61·9-s − 0.848·11-s − 5.89·12-s − 0.804·13-s + 4.75·14-s + 35/2·16-s − 0.642·17-s + 5.71·18-s + 0.707·19-s + 1.05·21-s + 3.00·22-s − 1.67·23-s + 9.73·24-s + 2.84·26-s + 1.22·27-s − 10.0·28-s + 1.37·29-s + 0.315·31-s − 22.2·32-s + 0.667·33-s + 2.27·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(26-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s+25/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(2^{5} \cdot 5^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.04304\times 10^{11}\) |
| Root analytic conductor: | \(14.0711\) |
| Motivic weight: | \(25\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((10,\ 2^{5} \cdot 5^{10} ,\ ( \ : 25/2, 25/2, 25/2, 25/2, 25/2 ),\ 1 )\) |
Particular Values
| \(L(13)\) | \(\approx\) | \(0.2472419395\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2472419395\) |
| \(L(\frac{27}{2})\) | 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 | $C_1$ | \( ( 1 + p^{12} T )^{5} \) |
| 5 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_5$ | \( 1 + 723995 T + 630872740975 p T^{2} + 5788990056353210 p^{5} T^{3} + \)\(11\!\cdots\!55\)\( p^{7} T^{4} + \)\(80\!\cdots\!03\)\( p^{13} T^{5} + \)\(11\!\cdots\!55\)\( p^{32} T^{6} + 5788990056353210 p^{55} T^{7} + 630872740975 p^{76} T^{8} + 723995 p^{100} T^{9} + p^{125} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + 49218886190 T + \)\(29\!\cdots\!25\)\( p T^{2} + \)\(25\!\cdots\!60\)\( p^{2} T^{3} + \)\(28\!\cdots\!30\)\( p^{5} T^{4} + \)\(20\!\cdots\!08\)\( p^{8} T^{5} + \)\(28\!\cdots\!30\)\( p^{30} T^{6} + \)\(25\!\cdots\!60\)\( p^{52} T^{7} + \)\(29\!\cdots\!25\)\( p^{76} T^{8} + 49218886190 p^{100} T^{9} + p^{125} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 + 8837033983815 T + \)\(70\!\cdots\!95\)\( p T^{2} + \)\(73\!\cdots\!30\)\( p^{2} T^{3} + \)\(12\!\cdots\!85\)\( p^{4} T^{4} + \)\(75\!\cdots\!93\)\( p^{6} T^{5} + \)\(12\!\cdots\!85\)\( p^{29} T^{6} + \)\(73\!\cdots\!30\)\( p^{52} T^{7} + \)\(70\!\cdots\!95\)\( p^{76} T^{8} + 8837033983815 p^{100} T^{9} + p^{125} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 67609989586220 T + \)\(10\!\cdots\!25\)\( T^{2} - \)\(62\!\cdots\!40\)\( p T^{3} + \)\(25\!\cdots\!90\)\( p^{2} T^{4} - \)\(12\!\cdots\!48\)\( p^{3} T^{5} + \)\(25\!\cdots\!90\)\( p^{27} T^{6} - \)\(62\!\cdots\!40\)\( p^{51} T^{7} + \)\(10\!\cdots\!25\)\( p^{75} T^{8} + 67609989586220 p^{100} T^{9} + p^{125} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 1543441123223415 T + \)\(10\!\cdots\!75\)\( T^{2} + \)\(96\!\cdots\!70\)\( p T^{3} + \)\(30\!\cdots\!65\)\( p^{2} T^{4} + \)\(19\!\cdots\!73\)\( p^{4} T^{5} + \)\(30\!\cdots\!65\)\( p^{27} T^{6} + \)\(96\!\cdots\!70\)\( p^{51} T^{7} + \)\(10\!\cdots\!75\)\( p^{75} T^{8} + 1543441123223415 p^{100} T^{9} + p^{125} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 - 6828205451077225 T + \)\(19\!\cdots\!55\)\( p T^{2} - \)\(11\!\cdots\!50\)\( p T^{3} + \)\(89\!\cdots\!15\)\( p^{3} T^{4} - \)\(40\!\cdots\!25\)\( p^{3} T^{5} + \)\(89\!\cdots\!15\)\( p^{28} T^{6} - \)\(11\!\cdots\!50\)\( p^{51} T^{7} + \)\(19\!\cdots\!55\)\( p^{76} T^{8} - 6828205451077225 p^{100} T^{9} + p^{125} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 + 7637191269884190 p T + \)\(88\!\cdots\!75\)\( p^{2} T^{2} + \)\(45\!\cdots\!40\)\( p^{3} T^{3} + \)\(32\!\cdots\!10\)\( p^{4} T^{4} + \)\(12\!\cdots\!08\)\( p^{5} T^{5} + \)\(32\!\cdots\!10\)\( p^{29} T^{6} + \)\(45\!\cdots\!40\)\( p^{53} T^{7} + \)\(88\!\cdots\!75\)\( p^{77} T^{8} + 7637191269884190 p^{101} T^{9} + p^{125} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 - 2616960879251719800 T + \)\(98\!\cdots\!45\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!10\)\( T^{4} - \)\(95\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!10\)\( p^{25} T^{6} - \)\(19\!\cdots\!00\)\( p^{50} T^{7} + \)\(98\!\cdots\!45\)\( p^{75} T^{8} - 2616960879251719800 p^{100} T^{9} + p^{125} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 - 1381921297247629210 T + \)\(44\!\cdots\!95\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(92\!\cdots\!10\)\( T^{4} + \)\(52\!\cdots\!48\)\( T^{5} + \)\(92\!\cdots\!10\)\( p^{25} T^{6} - \)\(13\!\cdots\!20\)\( p^{50} T^{7} + \)\(44\!\cdots\!95\)\( p^{75} T^{8} - 1381921297247629210 p^{100} T^{9} + p^{125} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 1839317412372824870 p T + \)\(56\!\cdots\!25\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} + \)\(40\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!10\)\( p^{25} T^{6} + \)\(21\!\cdots\!40\)\( p^{50} T^{7} + \)\(56\!\cdots\!25\)\( p^{75} T^{8} + 1839317412372824870 p^{101} T^{9} + p^{125} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 - \)\(11\!\cdots\!85\)\( T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(22\!\cdots\!70\)\( T^{3} + \)\(19\!\cdots\!85\)\( T^{4} + \)\(56\!\cdots\!73\)\( T^{5} + \)\(19\!\cdots\!85\)\( p^{25} T^{6} - \)\(22\!\cdots\!70\)\( p^{50} T^{7} + \)\(10\!\cdots\!95\)\( p^{75} T^{8} - \)\(11\!\cdots\!85\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 - \)\(16\!\cdots\!80\)\( T + \)\(22\!\cdots\!75\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!10\)\( T^{4} - \)\(26\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!10\)\( p^{25} T^{6} - \)\(28\!\cdots\!20\)\( p^{50} T^{7} + \)\(22\!\cdots\!75\)\( p^{75} T^{8} - \)\(16\!\cdots\!80\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 - \)\(16\!\cdots\!60\)\( T + \)\(28\!\cdots\!75\)\( T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!10\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(28\!\cdots\!10\)\( p^{25} T^{6} - \)\(27\!\cdots\!60\)\( p^{50} T^{7} + \)\(28\!\cdots\!75\)\( p^{75} T^{8} - \)\(16\!\cdots\!60\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 + \)\(42\!\cdots\!70\)\( T + \)\(39\!\cdots\!25\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(87\!\cdots\!10\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{5} + \)\(87\!\cdots\!10\)\( p^{25} T^{6} + \)\(15\!\cdots\!80\)\( p^{50} T^{7} + \)\(39\!\cdots\!25\)\( p^{75} T^{8} + \)\(42\!\cdots\!70\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(11\!\cdots\!95\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!90\)\( p T^{4} - \)\(65\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!90\)\( p^{26} T^{6} - \)\(23\!\cdots\!00\)\( p^{50} T^{7} + \)\(11\!\cdots\!95\)\( p^{75} T^{8} - \)\(33\!\cdots\!00\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + \)\(11\!\cdots\!90\)\( T + \)\(73\!\cdots\!45\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(57\!\cdots\!10\)\( T^{4} + \)\(68\!\cdots\!48\)\( T^{5} + \)\(57\!\cdots\!10\)\( p^{25} T^{6} + \)\(18\!\cdots\!80\)\( p^{50} T^{7} + \)\(73\!\cdots\!45\)\( p^{75} T^{8} + \)\(11\!\cdots\!90\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 - \)\(92\!\cdots\!35\)\( T + \)\(13\!\cdots\!25\)\( T^{2} - \)\(12\!\cdots\!10\)\( T^{3} + \)\(11\!\cdots\!85\)\( T^{4} - \)\(72\!\cdots\!17\)\( T^{5} + \)\(11\!\cdots\!85\)\( p^{25} T^{6} - \)\(12\!\cdots\!10\)\( p^{50} T^{7} + \)\(13\!\cdots\!25\)\( p^{75} T^{8} - \)\(92\!\cdots\!35\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 - \)\(17\!\cdots\!60\)\( T + \)\(54\!\cdots\!95\)\( T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!10\)\( p^{25} T^{6} - \)\(93\!\cdots\!20\)\( p^{50} T^{7} + \)\(54\!\cdots\!95\)\( p^{75} T^{8} - \)\(17\!\cdots\!60\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + \)\(19\!\cdots\!45\)\( T + \)\(14\!\cdots\!75\)\( T^{2} + \)\(29\!\cdots\!30\)\( T^{3} + \)\(97\!\cdots\!85\)\( T^{4} + \)\(16\!\cdots\!19\)\( T^{5} + \)\(97\!\cdots\!85\)\( p^{25} T^{6} + \)\(29\!\cdots\!30\)\( p^{50} T^{7} + \)\(14\!\cdots\!75\)\( p^{75} T^{8} + \)\(19\!\cdots\!45\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + \)\(22\!\cdots\!50\)\( T + \)\(87\!\cdots\!95\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} - \)\(25\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!10\)\( p^{25} T^{6} + \)\(11\!\cdots\!00\)\( p^{50} T^{7} + \)\(87\!\cdots\!95\)\( p^{75} T^{8} + \)\(22\!\cdots\!50\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + \)\(10\!\cdots\!45\)\( T + \)\(33\!\cdots\!25\)\( T^{2} + \)\(34\!\cdots\!30\)\( T^{3} + \)\(54\!\cdots\!85\)\( T^{4} + \)\(46\!\cdots\!19\)\( T^{5} + \)\(54\!\cdots\!85\)\( p^{25} T^{6} + \)\(34\!\cdots\!30\)\( p^{50} T^{7} + \)\(33\!\cdots\!25\)\( p^{75} T^{8} + \)\(10\!\cdots\!45\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 + \)\(59\!\cdots\!25\)\( T + \)\(28\!\cdots\!95\)\( T^{2} + \)\(98\!\cdots\!50\)\( T^{3} + \)\(30\!\cdots\!85\)\( T^{4} + \)\(73\!\cdots\!75\)\( T^{5} + \)\(30\!\cdots\!85\)\( p^{25} T^{6} + \)\(98\!\cdots\!50\)\( p^{50} T^{7} + \)\(28\!\cdots\!95\)\( p^{75} T^{8} + \)\(59\!\cdots\!25\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 - \)\(37\!\cdots\!10\)\( T + \)\(12\!\cdots\!25\)\( T^{2} - \)\(74\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!10\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + \)\(86\!\cdots\!10\)\( p^{25} T^{6} - \)\(74\!\cdots\!60\)\( p^{50} T^{7} + \)\(12\!\cdots\!25\)\( p^{75} T^{8} - \)\(37\!\cdots\!10\)\( p^{100} T^{9} + p^{125} T^{10} \) | |
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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
−5.89119984916218589800210601753, −5.48879488107897680359427254330, −5.42005569795753643309820262090, −5.40719265683779097234132018946, −5.24099133003850302459814458622, −4.65895881081937269319041791824, −4.13796511521541180040762241434, −4.05938487370558765999702305479, −3.80756611501412370073099391362, −3.57244058315239107294767432002, −2.94053348170299679686431244142, −2.85633216342282447349904007050, −2.76704271504475840933055190327, −2.70063988704325111666034282128, −2.53961888442457193747195524529, −2.17119819878770355508022833182, −1.72158782196861160034509911761, −1.68118916174707961119269965703, −1.55568218445186910296542538535, −1.13192203364311694255999075291, −0.71343175227787297074294208751, −0.57944673542836095061428808226, −0.49075519572031037821998914014, −0.25953383148419806882768795907, −0.25252090467895608243274312224, 0.25252090467895608243274312224, 0.25953383148419806882768795907, 0.49075519572031037821998914014, 0.57944673542836095061428808226, 0.71343175227787297074294208751, 1.13192203364311694255999075291, 1.55568218445186910296542538535, 1.68118916174707961119269965703, 1.72158782196861160034509911761, 2.17119819878770355508022833182, 2.53961888442457193747195524529, 2.70063988704325111666034282128, 2.76704271504475840933055190327, 2.85633216342282447349904007050, 2.94053348170299679686431244142, 3.57244058315239107294767432002, 3.80756611501412370073099391362, 4.05938487370558765999702305479, 4.13796511521541180040762241434, 4.65895881081937269319041791824, 5.24099133003850302459814458622, 5.40719265683779097234132018946, 5.42005569795753643309820262090, 5.48879488107897680359427254330, 5.89119984916218589800210601753