Dirichlet series
| L(s) = 1 | + 80·2-s + 81·3-s + 3.84e3·4-s + 1.21e3·5-s + 6.48e3·6-s − 8.71e3·7-s + 1.43e5·8-s − 3.36e4·9-s + 9.70e4·10-s + 4.39e3·11-s + 3.11e5·12-s − 6.97e5·14-s + 9.82e4·15-s + 4.58e6·16-s − 5.37e5·17-s − 2.69e6·18-s − 6.28e4·19-s + 4.65e6·20-s − 7.05e5·21-s + 3.51e5·22-s + 2.30e6·23-s + 1.16e7·24-s − 5.07e6·25-s − 2.12e6·27-s − 3.34e7·28-s − 4.74e6·29-s + 7.86e6·30-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 0.577·3-s + 15/2·4-s + 0.867·5-s + 2.04·6-s − 1.37·7-s + 12.3·8-s − 1.70·9-s + 3.06·10-s + 0.0904·11-s + 4.33·12-s − 4.85·14-s + 0.501·15-s + 35/2·16-s − 1.56·17-s − 6.04·18-s − 0.110·19-s + 6.50·20-s − 0.792·21-s + 0.319·22-s + 1.71·23-s + 7.14·24-s − 2.59·25-s − 0.770·27-s − 10.2·28-s − 1.24·29-s + 1.77·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(2^{5} \cdot 13^{10}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.59871\times 10^{11}\) |
| Root analytic conductor: | \(13.1940\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 2^{5} \cdot 13^{10} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )\) |
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 | 2 | $C_1$ | \( ( 1 - p^{4} T )^{5} \) |
| 13 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_5$ | \( 1 - p^{4} T + 40214 T^{2} - 1285097 p T^{3} + 124403833 p^{2} T^{4} - 4619356576 p^{3} T^{5} + 124403833 p^{11} T^{6} - 1285097 p^{19} T^{7} + 40214 p^{27} T^{8} - p^{40} T^{9} + p^{45} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 1213 T + 6546272 T^{2} - 1903960627 p T^{3} + 842667986683 p^{2} T^{4} - 218451868265964 p^{3} T^{5} + 842667986683 p^{11} T^{6} - 1903960627 p^{19} T^{7} + 6546272 p^{27} T^{8} - 1213 p^{36} T^{9} + p^{45} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 + 1245 p T + 17072290 p T^{2} + 135022585779 p T^{3} + 173665414340437 p^{2} T^{4} + 136233575908719564 p^{3} T^{5} + 173665414340437 p^{11} T^{6} + 135022585779 p^{19} T^{7} + 17072290 p^{28} T^{8} + 1245 p^{37} T^{9} + p^{45} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 - 4392 T + 4261822123 T^{2} + 65323412544672 T^{3} + 10299064607212501774 T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + 10299064607212501774 p^{9} T^{6} + 65323412544672 p^{18} T^{7} + 4261822123 p^{27} T^{8} - 4392 p^{36} T^{9} + p^{45} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 537741 T + 326634967512 T^{2} + 51292659965423007 T^{3} + \)\(13\!\cdots\!35\)\( T^{4} - \)\(34\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!35\)\( p^{9} T^{6} + 51292659965423007 p^{18} T^{7} + 326634967512 p^{27} T^{8} + 537741 p^{36} T^{9} + p^{45} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 62862 T + 836310412319 T^{2} + 206855644851250728 T^{3} + \)\(41\!\cdots\!66\)\( T^{4} + \)\(89\!\cdots\!60\)\( T^{5} + \)\(41\!\cdots\!66\)\( p^{9} T^{6} + 206855644851250728 p^{18} T^{7} + 836310412319 p^{27} T^{8} + 62862 p^{36} T^{9} + p^{45} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 - 100182 p T + 5773367228463 T^{2} - 8948191297138717104 T^{3} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!06\)\( p^{9} T^{6} - 8948191297138717104 p^{18} T^{7} + 5773367228463 p^{27} T^{8} - 100182 p^{37} T^{9} + p^{45} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 + 4749348 T + 51705259148277 T^{2} + \)\(18\!\cdots\!72\)\( T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + \)\(35\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!94\)\( p^{9} T^{6} + \)\(18\!\cdots\!72\)\( p^{18} T^{7} + 51705259148277 p^{27} T^{8} + 4749348 p^{36} T^{9} + p^{45} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 13160376 T + 130123705074703 T^{2} + \)\(94\!\cdots\!72\)\( T^{3} + \)\(58\!\cdots\!94\)\( T^{4} + \)\(31\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!94\)\( p^{9} T^{6} + \)\(94\!\cdots\!72\)\( p^{18} T^{7} + 130123705074703 p^{27} T^{8} + 13160376 p^{36} T^{9} + p^{45} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 19080099 T + 675452932258864 T^{2} + \)\(83\!\cdots\!85\)\( T^{3} + \)\(16\!\cdots\!91\)\( T^{4} + \)\(15\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!91\)\( p^{9} T^{6} + \)\(83\!\cdots\!85\)\( p^{18} T^{7} + 675452932258864 p^{27} T^{8} + 19080099 p^{36} T^{9} + p^{45} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 - 8434304 T + 1210135630652237 T^{2} - \)\(67\!\cdots\!76\)\( T^{3} + \)\(65\!\cdots\!66\)\( T^{4} - \)\(26\!\cdots\!04\)\( T^{5} + \)\(65\!\cdots\!66\)\( p^{9} T^{6} - \)\(67\!\cdots\!76\)\( p^{18} T^{7} + 1210135630652237 p^{27} T^{8} - 8434304 p^{36} T^{9} + p^{45} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 + 25704881 T + 1035047415552366 T^{2} + \)\(20\!\cdots\!83\)\( T^{3} + \)\(64\!\cdots\!85\)\( T^{4} + \)\(98\!\cdots\!00\)\( T^{5} + \)\(64\!\cdots\!85\)\( p^{9} T^{6} + \)\(20\!\cdots\!83\)\( p^{18} T^{7} + 1035047415552366 p^{27} T^{8} + 25704881 p^{36} T^{9} + p^{45} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 27047883 T + 3850344060140614 T^{2} + \)\(12\!\cdots\!61\)\( T^{3} + \)\(68\!\cdots\!37\)\( T^{4} + \)\(20\!\cdots\!04\)\( T^{5} + \)\(68\!\cdots\!37\)\( p^{9} T^{6} + \)\(12\!\cdots\!61\)\( p^{18} T^{7} + 3850344060140614 p^{27} T^{8} + 27047883 p^{36} T^{9} + p^{45} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 + 48759270 T + 9714179887292065 T^{2} + \)\(35\!\cdots\!24\)\( T^{3} + \)\(46\!\cdots\!18\)\( T^{4} + \)\(13\!\cdots\!24\)\( T^{5} + \)\(46\!\cdots\!18\)\( p^{9} T^{6} + \)\(35\!\cdots\!24\)\( p^{18} T^{7} + 9714179887292065 p^{27} T^{8} + 48759270 p^{36} T^{9} + p^{45} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - 171497614 T + 47573489055058631 T^{2} - \)\(56\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!78\)\( p T^{4} - \)\(72\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!78\)\( p^{10} T^{6} - \)\(56\!\cdots\!60\)\( p^{18} T^{7} + 47573489055058631 p^{27} T^{8} - 171497614 p^{36} T^{9} + p^{45} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 35079548 T + 40823354586252789 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(82\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(82\!\cdots\!74\)\( p^{9} T^{6} + \)\(15\!\cdots\!88\)\( p^{18} T^{7} + 40823354586252789 p^{27} T^{8} + 35079548 p^{36} T^{9} + p^{45} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 - 95187276 T + 80194520009101171 T^{2} - \)\(95\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!46\)\( T^{4} - \)\(37\!\cdots\!68\)\( T^{5} + \)\(31\!\cdots\!46\)\( p^{9} T^{6} - \)\(95\!\cdots\!12\)\( p^{18} T^{7} + 80194520009101171 p^{27} T^{8} - 95187276 p^{36} T^{9} + p^{45} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 140119209 T + 43026899192149270 T^{2} + \)\(37\!\cdots\!91\)\( T^{3} + \)\(36\!\cdots\!05\)\( T^{4} + \)\(70\!\cdots\!44\)\( T^{5} + \)\(36\!\cdots\!05\)\( p^{9} T^{6} + \)\(37\!\cdots\!91\)\( p^{18} T^{7} + 43026899192149270 p^{27} T^{8} + 140119209 p^{36} T^{9} + p^{45} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 332164038 T + 216292684358546953 T^{2} + \)\(45\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} + \)\(31\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!62\)\( p^{9} T^{6} + \)\(45\!\cdots\!92\)\( p^{18} T^{7} + 216292684358546953 p^{27} T^{8} + 332164038 p^{36} T^{9} + p^{45} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 - 318519174 T + 424685586634562087 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!34\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(92\!\cdots\!34\)\( p^{9} T^{6} - \)\(13\!\cdots\!40\)\( p^{18} T^{7} + 424685586634562087 p^{27} T^{8} - 318519174 p^{36} T^{9} + p^{45} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 623574146 T + 601317438035757503 T^{2} + \)\(24\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} + \)\(48\!\cdots\!20\)\( T^{5} + \)\(14\!\cdots\!06\)\( p^{9} T^{6} + \)\(24\!\cdots\!76\)\( p^{18} T^{7} + 601317438035757503 p^{27} T^{8} + 623574146 p^{36} T^{9} + p^{45} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 + 1327892442 T + 1073289086583458521 T^{2} + \)\(56\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!54\)\( T^{4} + \)\(15\!\cdots\!28\)\( T^{5} + \)\(31\!\cdots\!54\)\( p^{9} T^{6} + \)\(56\!\cdots\!68\)\( p^{18} T^{7} + 1073289086583458521 p^{27} T^{8} + 1327892442 p^{36} T^{9} + p^{45} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 1024350060 T + 2806224245051472725 T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(32\!\cdots\!14\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!14\)\( p^{9} T^{6} + \)\(19\!\cdots\!76\)\( p^{18} T^{7} + 2806224245051472725 p^{27} T^{8} + 1024350060 p^{36} T^{9} + p^{45} T^{10} \) | |
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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
−5.75575831144962386490457274547, −5.61750915012586520836880218465, −5.61594702849806079642856678280, −5.51304728128828451473532881047, −5.46837900496651823806315531616, −4.98333691217376005469740617209, −4.88797940629245906737170521623, −4.82475621696565376657921585013, −4.15497791443843685935129948590, −4.11947118373341757537485467031, −3.93873525965468524079836546889, −3.79671207916031095735720000261, −3.58078220769695803453140826629, −3.40174600400432903921925728661, −3.20422910674844229620880066043, −2.85226297050499863679895496632, −2.74368518291980501046776540379, −2.50733979190281655119094016248, −2.41491797185352455186151659211, −2.35420698457045473149727549633, −1.70032944961162772417050686588, −1.69823388249055958804210361290, −1.53505785195438377217324145449, −1.31462816037204565976767415558, −1.08876424531273946292476926076, 0, 0, 0, 0, 0, 1.08876424531273946292476926076, 1.31462816037204565976767415558, 1.53505785195438377217324145449, 1.69823388249055958804210361290, 1.70032944961162772417050686588, 2.35420698457045473149727549633, 2.41491797185352455186151659211, 2.50733979190281655119094016248, 2.74368518291980501046776540379, 2.85226297050499863679895496632, 3.20422910674844229620880066043, 3.40174600400432903921925728661, 3.58078220769695803453140826629, 3.79671207916031095735720000261, 3.93873525965468524079836546889, 4.11947118373341757537485467031, 4.15497791443843685935129948590, 4.82475621696565376657921585013, 4.88797940629245906737170521623, 4.98333691217376005469740617209, 5.46837900496651823806315531616, 5.51304728128828451473532881047, 5.61594702849806079642856678280, 5.61750915012586520836880218465, 5.75575831144962386490457274547