Dirichlet series
| L(s) = 1 | − 2.20e6·2-s + 2.44e10·3-s − 5.84e12·4-s + 5.35e14·5-s − 5.39e16·6-s + 3.01e17·7-s + 1.77e19·8-s + 4.40e19·9-s − 1.18e21·10-s + 2.66e22·11-s − 1.42e23·12-s + 2.67e24·13-s − 6.67e23·14-s + 1.30e25·15-s − 2.90e25·16-s − 4.08e26·17-s − 9.73e25·18-s + 1.57e27·19-s − 3.12e27·20-s + 7.36e27·21-s − 5.89e28·22-s − 1.21e27·23-s + 4.32e29·24-s − 2.71e30·25-s − 5.91e30·26-s − 2.22e30·27-s − 1.76e30·28-s + ⋯ |
| L(s) = 1 | − 0.745·2-s + 1.34·3-s − 0.664·4-s + 0.501·5-s − 1.00·6-s + 0.204·7-s + 0.678·8-s + 0.134·9-s − 0.374·10-s + 1.08·11-s − 0.894·12-s + 3.00·13-s − 0.152·14-s + 0.676·15-s − 0.375·16-s − 1.43·17-s − 0.100·18-s + 0.506·19-s − 0.333·20-s + 0.275·21-s − 0.809·22-s − 0.00642·23-s + 0.914·24-s − 2.39·25-s − 2.23·26-s − 0.374·27-s − 0.135·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(44-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+43/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1606.15\) |
| Root analytic conductor: | \(3.42213\) |
| Motivic weight: | \(43\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 1,\ (\ :43/2, 43/2, 43/2),\ 1)\) |
Particular Values
| \(L(22)\) | \(\approx\) | \(2.754586424\) |
| \(L(\frac12)\) | \(\approx\) | \(2.754586424\) |
| \(L(\frac{45}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| good | 2 | $S_4\times C_2$ | \( 1 + 276243 p^{3} T + 5236697541 p^{11} T^{2} + 4506292715385 p^{22} T^{3} + 5236697541 p^{54} T^{4} + 276243 p^{89} T^{5} + p^{129} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 903756956 p^{3} T + 9337321074916153 p^{10} T^{2} - \)\(87\!\cdots\!20\)\( p^{19} T^{3} + 9337321074916153 p^{53} T^{4} - 903756956 p^{89} T^{5} + p^{129} T^{6} \) | |
| 5 | $S_4\times C_2$ | \( 1 - 107041076154834 p T + \)\(48\!\cdots\!99\)\( p^{4} T^{2} - \)\(51\!\cdots\!56\)\( p^{9} T^{3} + \)\(48\!\cdots\!99\)\( p^{47} T^{4} - 107041076154834 p^{87} T^{5} + p^{129} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 - 6162682156438344 p^{2} T + \)\(95\!\cdots\!99\)\( p^{5} T^{2} - \)\(94\!\cdots\!00\)\( p^{9} T^{3} + \)\(95\!\cdots\!99\)\( p^{48} T^{4} - 6162682156438344 p^{88} T^{5} + p^{129} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 - \)\(24\!\cdots\!36\)\( p T + \)\(13\!\cdots\!65\)\( p^{2} T^{2} - \)\(17\!\cdots\!20\)\( p^{5} T^{3} + \)\(13\!\cdots\!65\)\( p^{45} T^{4} - \)\(24\!\cdots\!36\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!14\)\( p T + \)\(20\!\cdots\!71\)\( p^{3} T^{2} - \)\(98\!\cdots\!20\)\( p^{6} T^{3} + \)\(20\!\cdots\!71\)\( p^{46} T^{4} - \)\(20\!\cdots\!14\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!94\)\( T + \)\(98\!\cdots\!79\)\( p T^{2} + \)\(14\!\cdots\!40\)\( p^{3} T^{3} + \)\(98\!\cdots\!79\)\( p^{44} T^{4} + \)\(40\!\cdots\!94\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(83\!\cdots\!00\)\( p T + \)\(15\!\cdots\!03\)\( p^{3} T^{2} - \)\(21\!\cdots\!00\)\( p^{5} T^{3} + \)\(15\!\cdots\!03\)\( p^{46} T^{4} - \)\(83\!\cdots\!00\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(12\!\cdots\!48\)\( T + \)\(23\!\cdots\!99\)\( p T^{2} + \)\(70\!\cdots\!20\)\( p^{2} T^{3} + \)\(23\!\cdots\!99\)\( p^{44} T^{4} + \)\(12\!\cdots\!48\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(57\!\cdots\!50\)\( T + \)\(94\!\cdots\!23\)\( p T^{2} - \)\(95\!\cdots\!00\)\( p^{2} T^{3} + \)\(94\!\cdots\!23\)\( p^{44} T^{4} - \)\(57\!\cdots\!50\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(82\!\cdots\!04\)\( p T + \)\(55\!\cdots\!65\)\( p^{2} T^{2} + \)\(22\!\cdots\!80\)\( p^{3} T^{3} + \)\(55\!\cdots\!65\)\( p^{45} T^{4} + \)\(82\!\cdots\!04\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(62\!\cdots\!62\)\( p T + \)\(57\!\cdots\!47\)\( p^{2} T^{2} + \)\(24\!\cdots\!20\)\( p^{3} T^{3} + \)\(57\!\cdots\!47\)\( p^{45} T^{4} + \)\(62\!\cdots\!62\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(62\!\cdots\!26\)\( p T + \)\(15\!\cdots\!15\)\( p^{2} T^{2} - \)\(12\!\cdots\!20\)\( p^{3} T^{3} + \)\(15\!\cdots\!15\)\( p^{45} T^{4} - \)\(62\!\cdots\!26\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 - \)\(24\!\cdots\!92\)\( T + \)\(66\!\cdots\!57\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(66\!\cdots\!57\)\( p^{43} T^{4} - \)\(24\!\cdots\!92\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(30\!\cdots\!56\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!93\)\( p^{43} T^{4} - \)\(30\!\cdots\!56\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!62\)\( T + \)\(48\!\cdots\!47\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!47\)\( p^{43} T^{4} - \)\(15\!\cdots\!62\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(50\!\cdots\!37\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!37\)\( p^{43} T^{4} - \)\(22\!\cdots\!00\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(93\!\cdots\!54\)\( T + \)\(89\!\cdots\!15\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!15\)\( p^{43} T^{4} + \)\(93\!\cdots\!54\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(73\!\cdots\!44\)\( T + \)\(79\!\cdots\!93\)\( T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!93\)\( p^{43} T^{4} + \)\(73\!\cdots\!44\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!64\)\( T + \)\(22\!\cdots\!65\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!65\)\( p^{43} T^{4} + \)\(18\!\cdots\!64\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(20\!\cdots\!98\)\( T + \)\(46\!\cdots\!27\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!27\)\( p^{43} T^{4} + \)\(20\!\cdots\!98\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(11\!\cdots\!17\)\( T^{2} - \)\(15\!\cdots\!00\)\( p T^{3} + \)\(11\!\cdots\!17\)\( p^{43} T^{4} - \)\(15\!\cdots\!00\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(89\!\cdots\!28\)\( T + \)\(68\!\cdots\!17\)\( T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!17\)\( p^{43} T^{4} + \)\(89\!\cdots\!28\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!50\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!07\)\( p^{43} T^{4} - \)\(20\!\cdots\!50\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(38\!\cdots\!94\)\( T + \)\(63\!\cdots\!43\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!43\)\( p^{43} T^{4} + \)\(38\!\cdots\!94\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.41226615559705641608712516107, −18.23869234266349544116731822331, −17.72496964517227546683836679015, −17.70121612920923324856198162370, −16.09371509460355725212768295203, −15.85037283259849139834208795228, −14.64298428300446705778482044168, −13.90770841434162502850369730943, −13.69184573987314689104083966844, −13.14168844666834405967006159329, −11.62630428838795880795745896470, −11.02757837482997808670131795559, −9.992477227025999862014036048189, −8.881283421825448913641991162095, −8.878530349492648662914567154987, −8.611581403957278931214567809049, −7.46569587008960977917701953646, −6.26855815566352348958119860066, −5.74213419822693390400659725987, −4.04483222906678438653094116852, −4.00583767473095125197890886134, −2.93217208473476168019806100322, −1.93411219675686423614527871698, −1.36392634691248278259199015746, −0.52633746964524473091865152815, 0.52633746964524473091865152815, 1.36392634691248278259199015746, 1.93411219675686423614527871698, 2.93217208473476168019806100322, 4.00583767473095125197890886134, 4.04483222906678438653094116852, 5.74213419822693390400659725987, 6.26855815566352348958119860066, 7.46569587008960977917701953646, 8.611581403957278931214567809049, 8.878530349492648662914567154987, 8.881283421825448913641991162095, 9.992477227025999862014036048189, 11.02757837482997808670131795559, 11.62630428838795880795745896470, 13.14168844666834405967006159329, 13.69184573987314689104083966844, 13.90770841434162502850369730943, 14.64298428300446705778482044168, 15.85037283259849139834208795228, 16.09371509460355725212768295203, 17.70121612920923324856198162370, 17.72496964517227546683836679015, 18.23869234266349544116731822331, 19.41226615559705641608712516107