Dirichlet series
| L(s) = 1 | − 185·4-s + 266·5-s − 496·7-s − 474·8-s + 1.14e3·11-s − 642·13-s + 2.82e3·16-s + 5.79e3·17-s − 6.03e3·19-s − 4.92e4·20-s − 6.08e4·23-s − 2.91e5·25-s + 9.17e4·28-s + 1.69e5·29-s − 1.99e5·31-s + 1.81e5·32-s − 1.31e5·35-s − 2.02e5·37-s − 1.26e5·40-s − 5.41e5·41-s − 9.09e5·43-s − 2.12e5·44-s − 8.02e4·47-s − 1.51e6·49-s + 1.18e5·52-s + 2.78e5·53-s + 3.05e5·55-s + ⋯ |
| L(s) = 1 | − 1.44·4-s + 0.951·5-s − 0.546·7-s − 0.327·8-s + 0.260·11-s − 0.0810·13-s + 0.172·16-s + 0.286·17-s − 0.201·19-s − 1.37·20-s − 1.04·23-s − 3.72·25-s + 0.789·28-s + 1.28·29-s − 1.20·31-s + 0.976·32-s − 0.520·35-s − 0.655·37-s − 0.311·40-s − 1.22·41-s − 1.74·43-s − 0.375·44-s − 0.112·47-s − 1.83·49-s + 0.117·52-s + 0.256·53-s + 0.247·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(3^{10} \cdot 23^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.13058\times 10^{9}\) |
| Root analytic conductor: | \(8.04137\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 3^{10} \cdot 23^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | 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 | 3 | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 + p^{3} T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + 185 T^{2} + 237 p T^{3} + 7851 p^{2} T^{4} - 709 p^{3} T^{5} + 7851 p^{9} T^{6} + 237 p^{15} T^{7} + 185 p^{21} T^{8} + p^{35} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 266 T + 361929 T^{2} - 15472648 p T^{3} + 2161955522 p^{2} T^{4} - 14139712924 p^{4} T^{5} + 2161955522 p^{9} T^{6} - 15472648 p^{15} T^{7} + 361929 p^{21} T^{8} - 266 p^{28} T^{9} + p^{35} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 + 496 T + 1756571 T^{2} + 1123810704 T^{3} + 2372248497634 T^{4} + 984010614791616 T^{5} + 2372248497634 p^{7} T^{6} + 1123810704 p^{14} T^{7} + 1756571 p^{21} T^{8} + 496 p^{28} T^{9} + p^{35} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 - 1148 T + 79535271 T^{2} - 81036609776 T^{3} + 2759193912612338 T^{4} - 2286597617642676456 T^{5} + 2759193912612338 p^{7} T^{6} - 81036609776 p^{14} T^{7} + 79535271 p^{21} T^{8} - 1148 p^{28} T^{9} + p^{35} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 642 T + 107144993 T^{2} + 115024854072 T^{3} + 9280068844882210 T^{4} + 26329651650478741356 T^{5} + 9280068844882210 p^{7} T^{6} + 115024854072 p^{14} T^{7} + 107144993 p^{21} T^{8} + 642 p^{28} T^{9} + p^{35} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 - 5798 T + 765741381 T^{2} + 1663501787896 T^{3} + 370680130216694714 T^{4} + \)\(21\!\cdots\!20\)\( T^{5} + 370680130216694714 p^{7} T^{6} + 1663501787896 p^{14} T^{7} + 765741381 p^{21} T^{8} - 5798 p^{28} T^{9} + p^{35} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 6036 T + 3492386023 T^{2} + 25456470564896 T^{3} + 5555373680250650218 T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + 5555373680250650218 p^{7} T^{6} + 25456470564896 p^{14} T^{7} + 3492386023 p^{21} T^{8} + 6036 p^{28} T^{9} + p^{35} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 - 169162 T + 80325985009 T^{2} - 353217236227768 p T^{3} + \)\(26\!\cdots\!06\)\( T^{4} - \)\(25\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!06\)\( p^{7} T^{6} - 353217236227768 p^{15} T^{7} + 80325985009 p^{21} T^{8} - 169162 p^{28} T^{9} + p^{35} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 6440 p T + 3018327269 p T^{2} + 8543888532695840 T^{3} + \)\(29\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!94\)\( p^{7} T^{6} + 8543888532695840 p^{14} T^{7} + 3018327269 p^{22} T^{8} + 6440 p^{29} T^{9} + p^{35} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 202002 T + 369504819393 T^{2} + 64438923459378696 T^{3} + \)\(60\!\cdots\!22\)\( T^{4} + \)\(86\!\cdots\!00\)\( T^{5} + \)\(60\!\cdots\!22\)\( p^{7} T^{6} + 64438923459378696 p^{14} T^{7} + 369504819393 p^{21} T^{8} + 202002 p^{28} T^{9} + p^{35} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 322 p^{2} T + 870326731909 T^{2} + 355050034300082968 T^{3} + \)\(31\!\cdots\!30\)\( T^{4} + \)\(97\!\cdots\!60\)\( T^{5} + \)\(31\!\cdots\!30\)\( p^{7} T^{6} + 355050034300082968 p^{14} T^{7} + 870326731909 p^{21} T^{8} + 322 p^{30} T^{9} + p^{35} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 + 909596 T + 638610119087 T^{2} + 352087668310526336 T^{3} + \)\(24\!\cdots\!94\)\( T^{4} + \)\(28\!\cdots\!72\)\( p T^{5} + \)\(24\!\cdots\!94\)\( p^{7} T^{6} + 352087668310526336 p^{14} T^{7} + 638610119087 p^{21} T^{8} + 909596 p^{28} T^{9} + p^{35} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 80208 T + 1140209963019 T^{2} + 19449762264067776 T^{3} + \)\(89\!\cdots\!06\)\( T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!06\)\( p^{7} T^{6} + 19449762264067776 p^{14} T^{7} + 1140209963019 p^{21} T^{8} + 80208 p^{28} T^{9} + p^{35} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 - 278138 T + 2816263082553 T^{2} - 807494496605031560 T^{3} + \)\(38\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!56\)\( T^{5} + \)\(38\!\cdots\!78\)\( p^{7} T^{6} - 807494496605031560 p^{14} T^{7} + 2816263082553 p^{21} T^{8} - 278138 p^{28} T^{9} + p^{35} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 - 3177380 T + 9136544467935 T^{2} - 17964595219447971248 T^{3} + \)\(37\!\cdots\!66\)\( T^{4} - \)\(57\!\cdots\!28\)\( T^{5} + \)\(37\!\cdots\!66\)\( p^{7} T^{6} - 17964595219447971248 p^{14} T^{7} + 9136544467935 p^{21} T^{8} - 3177380 p^{28} T^{9} + p^{35} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 - 147782 T + 5932620594121 T^{2} - 9675976261596442392 T^{3} + \)\(88\!\cdots\!26\)\( T^{4} - \)\(58\!\cdots\!24\)\( T^{5} + \)\(88\!\cdots\!26\)\( p^{7} T^{6} - 9675976261596442392 p^{14} T^{7} + 5932620594121 p^{21} T^{8} - 147782 p^{28} T^{9} + p^{35} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 + 464916 T + 3276270412215 T^{2} + 10979955165261254592 T^{3} + \)\(60\!\cdots\!22\)\( T^{4} + \)\(18\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!22\)\( p^{7} T^{6} + 10979955165261254592 p^{14} T^{7} + 3276270412215 p^{21} T^{8} + 464916 p^{28} T^{9} + p^{35} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 1576792 T + 41978900215987 T^{2} + 57544448125800410272 T^{3} + \)\(73\!\cdots\!94\)\( T^{4} + \)\(78\!\cdots\!48\)\( T^{5} + \)\(73\!\cdots\!94\)\( p^{7} T^{6} + 57544448125800410272 p^{14} T^{7} + 41978900215987 p^{21} T^{8} + 1576792 p^{28} T^{9} + p^{35} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 38190 T + 47748528497349 T^{2} + 644610540830176872 T^{3} + \)\(97\!\cdots\!78\)\( T^{4} + \)\(61\!\cdots\!12\)\( T^{5} + \)\(97\!\cdots\!78\)\( p^{7} T^{6} + 644610540830176872 p^{14} T^{7} + 47748528497349 p^{21} T^{8} + 38190 p^{28} T^{9} + p^{35} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + 3913336 T + 76011562921523 T^{2} + \)\(29\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} + \)\(83\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!54\)\( p^{7} T^{6} + \)\(29\!\cdots\!68\)\( p^{14} T^{7} + 76011562921523 p^{21} T^{8} + 3913336 p^{28} T^{9} + p^{35} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 15774716 T + 194216440461567 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!42\)\( T^{4} + \)\(59\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!42\)\( p^{7} T^{6} + \)\(15\!\cdots\!20\)\( p^{14} T^{7} + 194216440461567 p^{21} T^{8} + 15774716 p^{28} T^{9} + p^{35} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 + 1116482 T + 104114520441501 T^{2} - \)\(20\!\cdots\!52\)\( T^{3} + \)\(49\!\cdots\!14\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!14\)\( p^{7} T^{6} - \)\(20\!\cdots\!52\)\( p^{14} T^{7} + 104114520441501 p^{21} T^{8} + 1116482 p^{28} T^{9} + p^{35} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 15738566 T + 381721971777741 T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!82\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(60\!\cdots\!82\)\( p^{7} T^{6} + \)\(44\!\cdots\!44\)\( p^{14} T^{7} + 381721971777741 p^{21} T^{8} + 15738566 p^{28} T^{9} + p^{35} T^{10} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91236919750194233849553592962, −6.81711031853431007319389295588, −6.32182602270299970830295828776, −6.30965392366812325065295618293, −6.07537980315793566553564130816, −5.90264535998772203988200689975, −5.39375865557537739891844954289, −5.25813660080119140097132036201, −5.24711781284460884863322312322, −5.23590192648070717499374856129, −4.49415137150874207805014125071, −4.39213204845544468683364100689, −4.08836457573597893513387823150, −3.96217389850277021219682826754, −3.86909662429958112936909752886, −3.37359029810187515472501938684, −3.30122256094787169411787415439, −2.90795986183609759987089929033, −2.46520344714925495768872060495, −2.38334305414622512033289138180, −2.01519326045091043598140080430, −1.81138324880307653563914988975, −1.30060382620511014207829667033, −1.30013875802404360395687104548, −1.11090079114978383101376851131, 0, 0, 0, 0, 0, 1.11090079114978383101376851131, 1.30013875802404360395687104548, 1.30060382620511014207829667033, 1.81138324880307653563914988975, 2.01519326045091043598140080430, 2.38334305414622512033289138180, 2.46520344714925495768872060495, 2.90795986183609759987089929033, 3.30122256094787169411787415439, 3.37359029810187515472501938684, 3.86909662429958112936909752886, 3.96217389850277021219682826754, 4.08836457573597893513387823150, 4.39213204845544468683364100689, 4.49415137150874207805014125071, 5.23590192648070717499374856129, 5.24711781284460884863322312322, 5.25813660080119140097132036201, 5.39375865557537739891844954289, 5.90264535998772203988200689975, 6.07537980315793566553564130816, 6.30965392366812325065295618293, 6.32182602270299970830295828776, 6.81711031853431007319389295588, 6.91236919750194233849553592962