Dirichlet series
| L(s) = 1 | + 2·2-s − 140·3-s − 862·4-s + 3.12e3·5-s − 280·6-s − 6.78e3·8-s − 5.50e3·9-s + 6.25e3·10-s + 1.03e4·11-s + 1.20e5·12-s − 1.58e5·13-s − 4.37e5·15-s + 1.23e5·16-s − 3.16e4·17-s − 1.10e4·18-s − 1.65e6·19-s − 2.69e6·20-s + 2.06e4·22-s + 7.96e5·23-s + 9.49e5·24-s + 5.85e6·25-s − 3.17e5·26-s + 1.20e6·27-s − 3.35e6·29-s − 8.75e5·30-s − 2.67e6·31-s + 6.04e6·32-s + ⋯ |
| L(s) = 1 | + 0.0883·2-s − 0.997·3-s − 1.68·4-s + 2.23·5-s − 0.0882·6-s − 0.585·8-s − 0.279·9-s + 0.197·10-s + 0.212·11-s + 1.68·12-s − 1.54·13-s − 2.23·15-s + 0.470·16-s − 0.0918·17-s − 0.0247·18-s − 2.91·19-s − 3.76·20-s + 0.0187·22-s + 0.593·23-s + 0.583·24-s + 3·25-s − 0.136·26-s + 0.437·27-s − 0.880·29-s − 0.197·30-s − 0.520·31-s + 1.01·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(5^{5} \cdot 7^{10}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.19902\times 10^{10}\) |
| Root analytic conductor: | \(11.2331\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 5^{5} \cdot 7^{10} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )\) |
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 | 5 | $C_1$ | \( ( 1 - p^{4} T )^{5} \) |
| 7 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_5$ | \( 1 - p T + 433 p T^{2} + 831 p^{2} T^{3} + 18845 p^{5} T^{4} + 27043 p^{6} T^{5} + 18845 p^{14} T^{6} + 831 p^{20} T^{7} + 433 p^{28} T^{8} - p^{37} T^{9} + p^{45} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 + 140 T + 25109 T^{2} + 3078964 T^{3} + 155837947 p T^{4} + 669598360 p^{2} T^{5} + 155837947 p^{10} T^{6} + 3078964 p^{18} T^{7} + 25109 p^{27} T^{8} + 140 p^{36} T^{9} + p^{45} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 - 10312 T + 6110501469 T^{2} + 12868286470036 T^{3} + 18721446338695361377 T^{4} + \)\(14\!\cdots\!88\)\( T^{5} + 18721446338695361377 p^{9} T^{6} + 12868286470036 p^{18} T^{7} + 6110501469 p^{27} T^{8} - 10312 p^{36} T^{9} + p^{45} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 158638 T + 40099411663 T^{2} + 4525202776113992 T^{3} + \)\(74\!\cdots\!09\)\( T^{4} + \)\(64\!\cdots\!38\)\( T^{5} + \)\(74\!\cdots\!09\)\( p^{9} T^{6} + 4525202776113992 p^{18} T^{7} + 40099411663 p^{27} T^{8} + 158638 p^{36} T^{9} + p^{45} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 31614 T + 152979949219 T^{2} + 15467440208130592 T^{3} + \)\(26\!\cdots\!33\)\( T^{4} + \)\(20\!\cdots\!30\)\( T^{5} + \)\(26\!\cdots\!33\)\( p^{9} T^{6} + 15467440208130592 p^{18} T^{7} + 152979949219 p^{27} T^{8} + 31614 p^{36} T^{9} + p^{45} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 1655376 T + 1555849159655 T^{2} + 44845180156018720 p T^{3} + \)\(28\!\cdots\!70\)\( T^{4} + \)\(95\!\cdots\!08\)\( T^{5} + \)\(28\!\cdots\!70\)\( p^{9} T^{6} + 44845180156018720 p^{19} T^{7} + 1555849159655 p^{27} T^{8} + 1655376 p^{36} T^{9} + p^{45} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 - 796104 T + 3980003153371 T^{2} - 4279841785977132736 T^{3} + \)\(51\!\cdots\!98\)\( p T^{4} - \)\(93\!\cdots\!40\)\( T^{5} + \)\(51\!\cdots\!98\)\( p^{10} T^{6} - 4279841785977132736 p^{18} T^{7} + 3980003153371 p^{27} T^{8} - 796104 p^{36} T^{9} + p^{45} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 + 3353726 T + 28116887828895 T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!45\)\( T^{4} + \)\(18\!\cdots\!78\)\( T^{5} + \)\(68\!\cdots\!45\)\( p^{9} T^{6} + \)\(11\!\cdots\!00\)\( p^{18} T^{7} + 28116887828895 p^{27} T^{8} + 3353726 p^{36} T^{9} + p^{45} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 2678120 T + 30980429107403 T^{2} + \)\(17\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!94\)\( T^{4} + \)\(47\!\cdots\!88\)\( T^{5} + \)\(13\!\cdots\!94\)\( p^{9} T^{6} + \)\(17\!\cdots\!84\)\( p^{18} T^{7} + 30980429107403 p^{27} T^{8} + 2678120 p^{36} T^{9} + p^{45} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 - 50994846 T + 39149870417693 p T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(49\!\cdots\!08\)\( T^{5} + \)\(40\!\cdots\!66\)\( p^{9} T^{6} - \)\(27\!\cdots\!92\)\( p^{18} T^{7} + 39149870417693 p^{28} T^{8} - 50994846 p^{36} T^{9} + p^{45} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 6330194 T + 1562261358649949 T^{2} + \)\(78\!\cdots\!48\)\( T^{3} + \)\(99\!\cdots\!30\)\( T^{4} + \)\(38\!\cdots\!76\)\( T^{5} + \)\(99\!\cdots\!30\)\( p^{9} T^{6} + \)\(78\!\cdots\!48\)\( p^{18} T^{7} + 1562261358649949 p^{27} T^{8} + 6330194 p^{36} T^{9} + p^{45} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 - 6149468 T + 2230226649262079 T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!34\)\( T^{4} - \)\(88\!\cdots\!36\)\( T^{5} + \)\(21\!\cdots\!34\)\( p^{9} T^{6} - \)\(11\!\cdots\!80\)\( p^{18} T^{7} + 2230226649262079 p^{27} T^{8} - 6149468 p^{36} T^{9} + p^{45} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 - 6897780 T + 2892726247790561 T^{2} + \)\(13\!\cdots\!96\)\( T^{3} + \)\(42\!\cdots\!25\)\( T^{4} + \)\(40\!\cdots\!32\)\( T^{5} + \)\(42\!\cdots\!25\)\( p^{9} T^{6} + \)\(13\!\cdots\!96\)\( p^{18} T^{7} + 2892726247790561 p^{27} T^{8} - 6897780 p^{36} T^{9} + p^{45} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 + 70886738 T + 8460080414958121 T^{2} - \)\(37\!\cdots\!40\)\( T^{3} - \)\(39\!\cdots\!94\)\( T^{4} - \)\(21\!\cdots\!32\)\( T^{5} - \)\(39\!\cdots\!94\)\( p^{9} T^{6} - \)\(37\!\cdots\!40\)\( p^{18} T^{7} + 8460080414958121 p^{27} T^{8} + 70886738 p^{36} T^{9} + p^{45} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - 2581952 p T + 34448863513048391 T^{2} - \)\(43\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!42\)\( T^{4} - \)\(51\!\cdots\!48\)\( T^{5} + \)\(52\!\cdots\!42\)\( p^{9} T^{6} - \)\(43\!\cdots\!68\)\( p^{18} T^{7} + 34448863513048391 p^{27} T^{8} - 2581952 p^{37} T^{9} + p^{45} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 257015698 T + 23075149340036769 T^{2} + \)\(27\!\cdots\!72\)\( T^{3} - \)\(13\!\cdots\!50\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} - \)\(13\!\cdots\!50\)\( p^{9} T^{6} + \)\(27\!\cdots\!72\)\( p^{18} T^{7} + 23075149340036769 p^{27} T^{8} + 257015698 p^{36} T^{9} + p^{45} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 - 133467828 T + 56930250312806863 T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!62\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!62\)\( p^{9} T^{6} + \)\(14\!\cdots\!40\)\( p^{18} T^{7} + 56930250312806863 p^{27} T^{8} - 133467828 p^{36} T^{9} + p^{45} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 - 522788960 T + 303569112441738147 T^{2} - \)\(93\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!82\)\( T^{4} - \)\(63\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!82\)\( p^{9} T^{6} - \)\(93\!\cdots\!00\)\( p^{18} T^{7} + 303569112441738147 p^{27} T^{8} - 522788960 p^{36} T^{9} + p^{45} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 159370858 T + 184580562002358805 T^{2} + \)\(29\!\cdots\!44\)\( T^{3} + \)\(17\!\cdots\!22\)\( T^{4} + \)\(23\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!22\)\( p^{9} T^{6} + \)\(29\!\cdots\!44\)\( p^{18} T^{7} + 184580562002358805 p^{27} T^{8} + 159370858 p^{36} T^{9} + p^{45} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 - 464174900 T + 596055521003027425 T^{2} - \)\(19\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!85\)\( T^{4} - \)\(33\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!85\)\( p^{9} T^{6} - \)\(19\!\cdots\!64\)\( p^{18} T^{7} + 596055521003027425 p^{27} T^{8} - 464174900 p^{36} T^{9} + p^{45} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 2207636832 T + 2812258639953749663 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} + \)\(76\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!98\)\( p^{9} T^{6} + \)\(24\!\cdots\!80\)\( p^{18} T^{7} + 2812258639953749663 p^{27} T^{8} + 2207636832 p^{36} T^{9} + p^{45} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 - 1106708326 T + 1689084835940499725 T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(68\!\cdots\!28\)\( T^{5} + \)\(11\!\cdots\!90\)\( p^{9} T^{6} - \)\(13\!\cdots\!80\)\( p^{18} T^{7} + 1689084835940499725 p^{27} T^{8} - 1106708326 p^{36} T^{9} + p^{45} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 1956142254 T + 4082161263087973027 T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!73\)\( T^{4} + \)\(57\!\cdots\!62\)\( T^{5} + \)\(63\!\cdots\!73\)\( p^{9} T^{6} + \)\(52\!\cdots\!20\)\( p^{18} T^{7} + 4082161263087973027 p^{27} T^{8} + 1956142254 p^{36} T^{9} + p^{45} T^{10} \) | |
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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
−6.25117477106111924586348974676, −6.17454509048884268283759516321, −5.85373021542288226424342112985, −5.79764574918453635171206269731, −5.74652874953702501568057093346, −5.14464262986209384490530581800, −5.01640389193112178800855375812, −4.97775391885228538037572759400, −4.96793185854812089187618134403, −4.41754284099383003188415755219, −4.27341014304543340790773275437, −4.10707807486108537811196238529, −3.90790378103958169411498975404, −3.78375957885512788358085952456, −3.13727928126702403109004778267, −2.80923187959128299378602079050, −2.70519464712728256680524099944, −2.48343376614821121640132260770, −2.31609737380688608516037658180, −2.16862709427524784189346127143, −1.92207085082649520950297427877, −1.30037993811704347638429323191, −1.18789691149311534645264530751, −1.10571582037988651838852286446, −0.890560454676593937200643856050, 0, 0, 0, 0, 0, 0.890560454676593937200643856050, 1.10571582037988651838852286446, 1.18789691149311534645264530751, 1.30037993811704347638429323191, 1.92207085082649520950297427877, 2.16862709427524784189346127143, 2.31609737380688608516037658180, 2.48343376614821121640132260770, 2.70519464712728256680524099944, 2.80923187959128299378602079050, 3.13727928126702403109004778267, 3.78375957885512788358085952456, 3.90790378103958169411498975404, 4.10707807486108537811196238529, 4.27341014304543340790773275437, 4.41754284099383003188415755219, 4.96793185854812089187618134403, 4.97775391885228538037572759400, 5.01640389193112178800855375812, 5.14464262986209384490530581800, 5.74652874953702501568057093346, 5.79764574918453635171206269731, 5.85373021542288226424342112985, 6.17454509048884268283759516321, 6.25117477106111924586348974676