Dirichlet series
| L(s) = 1 | + 694·5-s + 5.84e3·7-s − 2.91e4·9-s + 2.61e5·11-s + 1.48e6·13-s + 4.83e6·17-s − 2.76e6·19-s − 2.65e7·23-s − 2.95e7·25-s + 6.56e7·27-s + 1.14e8·29-s − 1.70e8·31-s + 4.05e6·35-s − 1.29e8·37-s − 6.72e7·41-s − 5.65e8·43-s − 2.02e7·45-s + 4.54e9·47-s − 5.52e9·49-s − 1.05e10·53-s + 1.81e8·55-s + 1.69e10·59-s − 3.60e9·61-s − 1.70e8·63-s + 1.03e9·65-s − 2.14e10·67-s − 5.90e10·71-s + ⋯ |
| L(s) = 1 | + 0.0993·5-s + 0.131·7-s − 0.164·9-s + 0.489·11-s + 1.10·13-s + 0.825·17-s − 0.255·19-s − 0.859·23-s − 0.604·25-s + 0.880·27-s + 1.03·29-s − 1.07·31-s + 0.0130·35-s − 0.306·37-s − 0.0906·41-s − 0.587·43-s − 0.0163·45-s + 2.89·47-s − 2.79·49-s − 3.45·53-s + 0.0486·55-s + 3.08·59-s − 0.546·61-s − 0.0216·63-s + 0.110·65-s − 1.93·67-s − 3.88·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{16} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.52340\times 10^{8}\) |
| Root analytic conductor: | \(12.6418\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{16} \cdot 13^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.06791240806\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06791240806\) |
| \(L(\frac{13}{2})\) | 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 | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - p^{5} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 29195 T^{2} - 2431664 p^{3} T^{3} + 99791152 p^{4} T^{4} - 2431664 p^{14} T^{5} + 29195 p^{22} T^{6} + p^{44} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 694 T + 30014169 T^{2} - 83335399158 p T^{3} - 36967951774916 p^{2} T^{4} - 83335399158 p^{12} T^{5} + 30014169 p^{22} T^{6} - 694 p^{33} T^{7} + p^{44} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - 5844 T + 794100701 p T^{2} - 12380001376488 p T^{3} + 282625238599087896 p^{2} T^{4} - 12380001376488 p^{12} T^{5} + 794100701 p^{23} T^{6} - 5844 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 261644 T + 33768117096 p T^{2} - 32893942140521244 T^{3} + \)\(14\!\cdots\!90\)\( T^{4} - 32893942140521244 p^{11} T^{5} + 33768117096 p^{23} T^{6} - 261644 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 4832302 T + 43228144520697 T^{2} - \)\(33\!\cdots\!46\)\( T^{3} + \)\(84\!\cdots\!08\)\( T^{4} - \)\(33\!\cdots\!46\)\( p^{11} T^{5} + 43228144520697 p^{22} T^{6} - 4832302 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 2762436 T + 236876206928456 T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!86\)\( T^{4} + \)\(11\!\cdots\!40\)\( p^{11} T^{5} + 236876206928456 p^{22} T^{6} + 2762436 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 26525496 T + 2547764319976028 T^{2} + \)\(60\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!98\)\( T^{4} + \)\(60\!\cdots\!72\)\( p^{11} T^{5} + 2547764319976028 p^{22} T^{6} + 26525496 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 114398784 T + 52507122978114236 T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(98\!\cdots\!86\)\( T^{4} - \)\(42\!\cdots\!40\)\( p^{11} T^{5} + 52507122978114236 p^{22} T^{6} - 114398784 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 170870588 T + 104753320924601356 T^{2} + \)\(12\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!24\)\( p^{11} T^{5} + 104753320924601356 p^{22} T^{6} + 170870588 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 129457314 T + 12232594027519277 p T^{2} + \)\(95\!\cdots\!02\)\( T^{3} + \)\(94\!\cdots\!48\)\( T^{4} + \)\(95\!\cdots\!02\)\( p^{11} T^{5} + 12232594027519277 p^{23} T^{6} + 129457314 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 67263908 T + 970526216755255464 T^{2} - \)\(41\!\cdots\!96\)\( T^{3} + \)\(41\!\cdots\!86\)\( T^{4} - \)\(41\!\cdots\!96\)\( p^{11} T^{5} + 970526216755255464 p^{22} T^{6} + 67263908 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 565894584 T + 1761263157683945243 T^{2} + \)\(81\!\cdots\!36\)\( T^{3} + \)\(23\!\cdots\!04\)\( T^{4} + \)\(81\!\cdots\!36\)\( p^{11} T^{5} + 1761263157683945243 p^{22} T^{6} + 565894584 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 4547521124 T + 17123384412073678275 T^{2} - \)\(38\!\cdots\!04\)\( T^{3} + \)\(73\!\cdots\!56\)\( T^{4} - \)\(38\!\cdots\!04\)\( p^{11} T^{5} + 17123384412073678275 p^{22} T^{6} - 4547521124 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 10533452684 T + 72581552647205242968 T^{2} + \)\(33\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!78\)\( T^{4} + \)\(33\!\cdots\!28\)\( p^{11} T^{5} + 72581552647205242968 p^{22} T^{6} + 10533452684 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 16921519420 T + \)\(21\!\cdots\!28\)\( T^{2} - \)\(17\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!62\)\( T^{4} - \)\(17\!\cdots\!88\)\( p^{11} T^{5} + \)\(21\!\cdots\!28\)\( p^{22} T^{6} - 16921519420 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 3601744572 T + \)\(12\!\cdots\!76\)\( T^{2} + \)\(37\!\cdots\!08\)\( T^{3} + \)\(77\!\cdots\!06\)\( T^{4} + \)\(37\!\cdots\!08\)\( p^{11} T^{5} + \)\(12\!\cdots\!76\)\( p^{22} T^{6} + 3601744572 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 21420671204 T + \)\(52\!\cdots\!20\)\( T^{2} + \)\(67\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!86\)\( p T^{4} + \)\(67\!\cdots\!76\)\( p^{11} T^{5} + \)\(52\!\cdots\!20\)\( p^{22} T^{6} + 21420671204 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 59097402980 T + \)\(20\!\cdots\!95\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!08\)\( T^{4} + \)\(49\!\cdots\!60\)\( p^{11} T^{5} + \)\(20\!\cdots\!95\)\( p^{22} T^{6} + 59097402980 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 16938424496 T + \)\(97\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!04\)\( T^{3} + \)\(44\!\cdots\!82\)\( T^{4} - \)\(12\!\cdots\!04\)\( p^{11} T^{5} + \)\(97\!\cdots\!20\)\( p^{22} T^{6} - 16938424496 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 38736725720 T + \)\(24\!\cdots\!88\)\( T^{2} + \)\(67\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!62\)\( T^{4} + \)\(67\!\cdots\!88\)\( p^{11} T^{5} + \)\(24\!\cdots\!88\)\( p^{22} T^{6} + 38736725720 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 67937054892 T + \)\(44\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(63\!\cdots\!62\)\( T^{4} - \)\(14\!\cdots\!12\)\( p^{11} T^{5} + \)\(44\!\cdots\!80\)\( p^{22} T^{6} - 67937054892 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 34556088472 T + \)\(16\!\cdots\!52\)\( T^{2} - \)\(41\!\cdots\!84\)\( T^{3} + \)\(54\!\cdots\!14\)\( T^{4} - \)\(41\!\cdots\!84\)\( p^{11} T^{5} + \)\(16\!\cdots\!52\)\( p^{22} T^{6} - 34556088472 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 75994027560 T + \)\(17\!\cdots\!16\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!30\)\( T^{4} - \)\(12\!\cdots\!12\)\( p^{11} T^{5} + \)\(17\!\cdots\!16\)\( p^{22} T^{6} - 75994027560 p^{33} T^{7} + p^{44} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.01240643814882706208231982848, −6.76483859842080419176798363407, −6.32916789129450198673808361254, −6.25644096989679109043066410705, −6.16744209789026247085413419388, −5.60733141319306863333786324562, −5.46850257839452478992657143464, −5.40643218573415550923132632931, −4.73652503018553585711565504740, −4.49680925809043192812896398008, −4.49074251656442968856820595871, −3.96393888452654199312078853898, −3.89926712722344449335340656521, −3.30809107648096187576005834171, −3.16433179452273064135930613017, −2.99655940457458270250280155929, −2.82356291509948311380713082362, −2.05326785309862074415469412062, −1.88386711158974862420080097229, −1.71639713679691997062016730540, −1.54880317823825209111943126163, −1.02699672363851337756101395871, −0.72390492371423954061207244168, −0.62930290181809996132556785858, −0.02290689091993670902520479769, 0.02290689091993670902520479769, 0.62930290181809996132556785858, 0.72390492371423954061207244168, 1.02699672363851337756101395871, 1.54880317823825209111943126163, 1.71639713679691997062016730540, 1.88386711158974862420080097229, 2.05326785309862074415469412062, 2.82356291509948311380713082362, 2.99655940457458270250280155929, 3.16433179452273064135930613017, 3.30809107648096187576005834171, 3.89926712722344449335340656521, 3.96393888452654199312078853898, 4.49074251656442968856820595871, 4.49680925809043192812896398008, 4.73652503018553585711565504740, 5.40643218573415550923132632931, 5.46850257839452478992657143464, 5.60733141319306863333786324562, 6.16744209789026247085413419388, 6.25644096989679109043066410705, 6.32916789129450198673808361254, 6.76483859842080419176798363407, 7.01240643814882706208231982848