Dirichlet series
| L(s) = 1 | + 33·2-s + 324·3-s − 309·4-s − 2.47e3·5-s + 1.06e4·6-s − 1.80e4·8-s + 6.56e4·9-s − 8.17e4·10-s − 2.67e4·11-s − 1.00e5·12-s − 1.40e5·13-s − 8.02e5·15-s + 1.44e5·16-s − 5.43e5·17-s + 2.16e6·18-s − 2.09e5·19-s + 7.65e5·20-s − 8.84e5·22-s − 6.66e4·23-s − 5.84e6·24-s + 1.22e6·25-s − 4.63e6·26-s + 1.06e7·27-s − 6.21e6·29-s − 2.64e7·30-s + 6.91e6·31-s + 6.28e6·32-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 2.30·3-s − 0.603·4-s − 1.77·5-s + 3.36·6-s − 1.55·8-s + 10/3·9-s − 2.58·10-s − 0.551·11-s − 1.39·12-s − 1.36·13-s − 4.09·15-s + 0.553·16-s − 1.57·17-s + 4.86·18-s − 0.368·19-s + 1.07·20-s − 0.804·22-s − 0.0496·23-s − 3.59·24-s + 0.627·25-s − 1.98·26-s + 3.84·27-s − 1.63·29-s − 5.97·30-s + 1.34·31-s + 1.06·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(3^{4} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.28563\times 10^{7}\) |
| Root analytic conductor: | \(8.70116\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 3^{4} \cdot 7^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 7 | \( 1 \) | ||
| good | 2 | $C_2 \wr S_4$ | \( 1 - 33 T + 699 p T^{2} - 4785 p^{3} T^{3} + 14915 p^{6} T^{4} - 4785 p^{12} T^{5} + 699 p^{19} T^{6} - 33 p^{27} T^{7} + p^{36} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 2478 T + 4915076 T^{2} + 1767824874 p T^{3} + 548757512934 p^{2} T^{4} + 1767824874 p^{10} T^{5} + 4915076 p^{18} T^{6} + 2478 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 26790 T + 3596911116 T^{2} - 2722433252730 T^{3} + 5243496024573674390 T^{4} - 2722433252730 p^{9} T^{5} + 3596911116 p^{18} T^{6} + 26790 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 140448 T + 27945555988 T^{2} + 1889422158337632 T^{3} + \)\(29\!\cdots\!74\)\( T^{4} + 1889422158337632 p^{9} T^{5} + 27945555988 p^{18} T^{6} + 140448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 31974 p T + 505633657820 T^{2} + 171799734488314482 T^{3} + \)\(88\!\cdots\!30\)\( T^{4} + 171799734488314482 p^{9} T^{5} + 505633657820 p^{18} T^{6} + 31974 p^{28} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 209220 T + 488828261020 T^{2} + 201514214834402148 T^{3} + \)\(14\!\cdots\!62\)\( T^{4} + 201514214834402148 p^{9} T^{5} + 488828261020 p^{18} T^{6} + 209220 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 66690 T + 2662421726052 T^{2} + 631844376279579018 T^{3} + \)\(43\!\cdots\!42\)\( T^{4} + 631844376279579018 p^{9} T^{5} + 2662421726052 p^{18} T^{6} + 66690 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 6216804 T + 61601625852804 T^{2} + \)\(24\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!76\)\( p^{9} T^{5} + 61601625852804 p^{18} T^{6} + 6216804 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 6910908 T + 67873240229404 T^{2} - \)\(41\!\cdots\!84\)\( T^{3} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(41\!\cdots\!84\)\( p^{9} T^{5} + 67873240229404 p^{18} T^{6} - 6910908 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 2536116 T + 113513553703444 T^{2} - \)\(16\!\cdots\!64\)\( T^{3} + \)\(51\!\cdots\!30\)\( p T^{4} - \)\(16\!\cdots\!64\)\( p^{9} T^{5} + 113513553703444 p^{18} T^{6} - 2536116 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 21866058 T + 1121211295415708 T^{2} + \)\(19\!\cdots\!74\)\( T^{3} + \)\(52\!\cdots\!14\)\( T^{4} + \)\(19\!\cdots\!74\)\( p^{9} T^{5} + 1121211295415708 p^{18} T^{6} + 21866058 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 57993648 T + 2589989503256716 T^{2} + \)\(78\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} + \)\(78\!\cdots\!24\)\( p^{9} T^{5} + 2589989503256716 p^{18} T^{6} + 57993648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 23771568 T + 2385535590461372 T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(25\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!32\)\( p^{9} T^{5} + 2385535590461372 p^{18} T^{6} - 23771568 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 55156332 T + 13630836319010916 T^{2} + \)\(54\!\cdots\!24\)\( T^{3} + \)\(68\!\cdots\!06\)\( T^{4} + \)\(54\!\cdots\!24\)\( p^{9} T^{5} + 13630836319010916 p^{18} T^{6} + 55156332 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 99247488 T + 16672464730608908 T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(86\!\cdots\!00\)\( p^{9} T^{5} + 16672464730608908 p^{18} T^{6} + 99247488 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 159659844 T + 9874618289619268 T^{2} - \)\(18\!\cdots\!48\)\( T^{3} - \)\(30\!\cdots\!86\)\( T^{4} - \)\(18\!\cdots\!48\)\( p^{9} T^{5} + 9874618289619268 p^{18} T^{6} + 159659844 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 161367812 T + 107071627121487964 T^{2} + \)\(12\!\cdots\!52\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(12\!\cdots\!52\)\( p^{9} T^{5} + 107071627121487964 p^{18} T^{6} + 161367812 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 404129562 T + 193250917591204692 T^{2} - \)\(55\!\cdots\!90\)\( T^{3} + \)\(13\!\cdots\!34\)\( T^{4} - \)\(55\!\cdots\!90\)\( p^{9} T^{5} + 193250917591204692 p^{18} T^{6} - 404129562 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 833772468 T + 482121258084478660 T^{2} + \)\(17\!\cdots\!32\)\( T^{3} + \)\(51\!\cdots\!62\)\( T^{4} + \)\(17\!\cdots\!32\)\( p^{9} T^{5} + 482121258084478660 p^{18} T^{6} + 833772468 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 169919972 T + 315689603487055228 T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(49\!\cdots\!06\)\( T^{4} - \)\(34\!\cdots\!60\)\( p^{9} T^{5} + 315689603487055228 p^{18} T^{6} - 169919972 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 1291623648 T + 1217953859247250412 T^{2} + \)\(75\!\cdots\!24\)\( T^{3} + \)\(38\!\cdots\!10\)\( T^{4} + \)\(75\!\cdots\!24\)\( p^{9} T^{5} + 1217953859247250412 p^{18} T^{6} + 1291623648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1604461830 T + 2210429871786311276 T^{2} - \)\(18\!\cdots\!66\)\( T^{3} + \)\(13\!\cdots\!34\)\( T^{4} - \)\(18\!\cdots\!66\)\( p^{9} T^{5} + 2210429871786311276 p^{18} T^{6} - 1604461830 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 2179753836 T + 4115476491012023044 T^{2} + \)\(48\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!50\)\( T^{4} + \)\(48\!\cdots\!84\)\( p^{9} T^{5} + 4115476491012023044 p^{18} T^{6} + 2179753836 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−8.281798777420412354735946331173, −8.008216121859671671352555562108, −7.81941455873314354378972205220, −7.69630723551414490253253479230, −7.48282873876711403201606874888, −7.05876006898933683385039817038, −6.81630403225314800704442486967, −6.40239335825011395230313984315, −6.23708889440144532037969775033, −5.41718244128961722190191319511, −5.30234085057953412630112134937, −4.89275508494528864203969604565, −4.80472618674293450059933635499, −4.36920025569243330555594915746, −4.09535422704396007436238728974, −4.03914917799135680587826125682, −4.00031622451347429214165252586, −3.25475421962308182566890335056, −3.17029488565312111109469731801, −2.98732824858160184655136516067, −2.38946847731581775870835617183, −2.32991761727483931171979915252, −1.67681694533080318389821068394, −1.46875458028025330270706654149, −1.12326152729935663265552560409, 0, 0, 0, 0, 1.12326152729935663265552560409, 1.46875458028025330270706654149, 1.67681694533080318389821068394, 2.32991761727483931171979915252, 2.38946847731581775870835617183, 2.98732824858160184655136516067, 3.17029488565312111109469731801, 3.25475421962308182566890335056, 4.00031622451347429214165252586, 4.03914917799135680587826125682, 4.09535422704396007436238728974, 4.36920025569243330555594915746, 4.80472618674293450059933635499, 4.89275508494528864203969604565, 5.30234085057953412630112134937, 5.41718244128961722190191319511, 6.23708889440144532037969775033, 6.40239335825011395230313984315, 6.81630403225314800704442486967, 7.05876006898933683385039817038, 7.48282873876711403201606874888, 7.69630723551414490253253479230, 7.81941455873314354378972205220, 8.008216121859671671352555562108, 8.281798777420412354735946331173