Dirichlet series
| L(s) = 1 | + 256·2-s + 182·3-s + 4.09e4·4-s + 6.44e4·5-s + 4.65e4·6-s + 5.24e6·8-s − 1.18e6·9-s + 1.64e7·10-s + 1.00e6·11-s + 7.45e6·12-s + 2.69e7·13-s + 1.17e7·15-s + 5.87e8·16-s + 1.65e8·17-s − 3.02e8·18-s + 4.23e8·19-s + 2.63e9·20-s + 2.58e8·22-s − 2.86e8·23-s + 9.54e8·24-s − 1.31e8·25-s + 6.88e9·26-s − 2.04e9·27-s + 9.10e9·29-s + 3.00e9·30-s − 1.50e9·31-s + 6.01e10·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.144·3-s + 5·4-s + 1.84·5-s + 0.407·6-s + 7.07·8-s − 0.740·9-s + 5.21·10-s + 0.171·11-s + 0.720·12-s + 1.54·13-s + 0.265·15-s + 35/4·16-s + 1.66·17-s − 2.09·18-s + 2.06·19-s + 9.21·20-s + 0.485·22-s − 0.403·23-s + 1.01·24-s − 0.107·25-s + 4.37·26-s − 1.01·27-s + 2.84·29-s + 0.751·30-s − 0.304·31-s + 9.89·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(92236816\) = \(2^{4} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.21950\times 10^{8}\) |
| Root analytic conductor: | \(10.2511\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 92236816,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(264.9807143\) |
| \(L(\frac12)\) | \(\approx\) | \(264.9807143\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) |
| 7 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_4$ | \( 1 - 182 T + 404444 p T^{2} + 59712184 p^{3} T^{3} + 2736578221 p^{6} T^{4} + 59712184 p^{16} T^{5} + 404444 p^{27} T^{6} - 182 p^{39} T^{7} + p^{52} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2576 p^{2} T + 4279077534 T^{2} - 36792035493192 p T^{3} + 61878878677386227 p^{3} T^{4} - 36792035493192 p^{14} T^{5} + 4279077534 p^{26} T^{6} - 2576 p^{41} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 1008790 T + 9097631660652 p T^{2} - 1595518238719660296 p^{2} T^{3} + \)\(33\!\cdots\!87\)\( p^{3} T^{4} - 1595518238719660296 p^{15} T^{5} + 9097631660652 p^{27} T^{6} - 1008790 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 26903632 T + 97747020241532 p T^{2} - \)\(22\!\cdots\!96\)\( T^{3} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!96\)\( p^{13} T^{5} + 97747020241532 p^{27} T^{6} - 26903632 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 165333028 T + 13033909201036578 T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(19\!\cdots\!39\)\( T^{4} - \)\(13\!\cdots\!04\)\( p^{13} T^{5} + 13033909201036578 p^{26} T^{6} - 165333028 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 423405794 T + 173792406871110052 T^{2} - \)\(42\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!09\)\( T^{4} - \)\(42\!\cdots\!12\)\( p^{13} T^{5} + 173792406871110052 p^{26} T^{6} - 423405794 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 286233866 T + 963119177685129240 T^{2} + \)\(13\!\cdots\!88\)\( T^{3} + \)\(50\!\cdots\!53\)\( T^{4} + \)\(13\!\cdots\!88\)\( p^{13} T^{5} + 963119177685129240 p^{26} T^{6} + 286233866 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 9100337408 T + 63557643354325261932 T^{2} - \)\(29\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(29\!\cdots\!08\)\( p^{13} T^{5} + 63557643354325261932 p^{26} T^{6} - 9100337408 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 1507094246 T + 87176381526328509416 T^{2} + \)\(93\!\cdots\!96\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} + \)\(93\!\cdots\!96\)\( p^{13} T^{5} + 87176381526328509416 p^{26} T^{6} + 1507094246 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 18959705336 T + \)\(74\!\cdots\!46\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!87\)\( T^{4} - \)\(80\!\cdots\!80\)\( p^{13} T^{5} + \)\(74\!\cdots\!46\)\( p^{26} T^{6} - 18959705336 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 56825955312 T + \)\(21\!\cdots\!32\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} - \)\(40\!\cdots\!08\)\( p^{13} T^{5} + \)\(21\!\cdots\!32\)\( p^{26} T^{6} - 56825955312 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 45778809712 T + \)\(48\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} + \)\(17\!\cdots\!40\)\( p^{13} T^{5} + \)\(48\!\cdots\!88\)\( p^{26} T^{6} + 45778809712 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 81351201078 T + \)\(12\!\cdots\!68\)\( T^{2} - \)\(21\!\cdots\!24\)\( T^{3} + \)\(47\!\cdots\!69\)\( T^{4} - \)\(21\!\cdots\!24\)\( p^{13} T^{5} + \)\(12\!\cdots\!68\)\( p^{26} T^{6} - 81351201078 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 87497947440 T + \)\(33\!\cdots\!70\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(35\!\cdots\!80\)\( p^{13} T^{5} + \)\(33\!\cdots\!70\)\( p^{26} T^{6} + 87497947440 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 194140265102 T + \)\(35\!\cdots\!80\)\( T^{2} - \)\(54\!\cdots\!52\)\( T^{3} + \)\(53\!\cdots\!33\)\( T^{4} - \)\(54\!\cdots\!52\)\( p^{13} T^{5} + \)\(35\!\cdots\!80\)\( p^{26} T^{6} - 194140265102 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 175816313120 T + \)\(61\!\cdots\!06\)\( p T^{2} + \)\(55\!\cdots\!72\)\( T^{3} + \)\(70\!\cdots\!95\)\( T^{4} + \)\(55\!\cdots\!72\)\( p^{13} T^{5} + \)\(61\!\cdots\!06\)\( p^{27} T^{6} + 175816313120 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 243815218758 T + \)\(13\!\cdots\!84\)\( T^{2} - \)\(22\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!93\)\( T^{4} - \)\(22\!\cdots\!56\)\( p^{13} T^{5} + \)\(13\!\cdots\!84\)\( p^{26} T^{6} - 243815218758 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 1637697339536 T + \)\(40\!\cdots\!88\)\( T^{2} + \)\(37\!\cdots\!32\)\( T^{3} + \)\(59\!\cdots\!54\)\( T^{4} + \)\(37\!\cdots\!32\)\( p^{13} T^{5} + \)\(40\!\cdots\!88\)\( p^{26} T^{6} + 1637697339536 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 3492491920596 T + \)\(92\!\cdots\!26\)\( T^{2} - \)\(16\!\cdots\!08\)\( T^{3} + \)\(25\!\cdots\!43\)\( T^{4} - \)\(16\!\cdots\!08\)\( p^{13} T^{5} + \)\(92\!\cdots\!26\)\( p^{26} T^{6} - 3492491920596 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1016380081246 T + \)\(28\!\cdots\!24\)\( T^{2} - \)\(42\!\cdots\!32\)\( T^{3} + \)\(41\!\cdots\!05\)\( T^{4} - \)\(42\!\cdots\!32\)\( p^{13} T^{5} + \)\(28\!\cdots\!24\)\( p^{26} T^{6} - 1016380081246 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 3513747871648 T + \)\(13\!\cdots\!36\)\( T^{2} - \)\(57\!\cdots\!88\)\( p T^{3} + \)\(22\!\cdots\!98\)\( T^{4} - \)\(57\!\cdots\!88\)\( p^{14} T^{5} + \)\(13\!\cdots\!36\)\( p^{26} T^{6} - 3513747871648 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 8034124428036 T + \)\(56\!\cdots\!62\)\( T^{2} - \)\(15\!\cdots\!32\)\( p T^{3} + \)\(72\!\cdots\!79\)\( T^{4} - \)\(15\!\cdots\!32\)\( p^{14} T^{5} + \)\(56\!\cdots\!62\)\( p^{26} T^{6} - 8034124428036 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 27175565862816 T + \)\(47\!\cdots\!36\)\( T^{2} - \)\(58\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(58\!\cdots\!56\)\( p^{13} T^{5} + \)\(47\!\cdots\!36\)\( p^{26} T^{6} - 27175565862816 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−7.67229086395533399586817564586, −7.35551899962927700637081966572, −6.68396746990042997226212728611, −6.54287880014927799329517777402, −6.37692104662818563199692830804, −5.99189146442465318952986336555, −5.69787892197818586327323992626, −5.64065692803407489379996253823, −5.60505819083254379737755408800, −5.14271291967608554379986689233, −4.70962905301239260105980502231, −4.44014614456768203370353940788, −4.27436398000170396485326690680, −3.58644321631515063876427990820, −3.41509312149737191717507938093, −3.25549948398977274777387431850, −3.20573841550072706773108418718, −2.53774997159346085651828387299, −2.22556888988327126909969840085, −2.16932305269839800279301302669, −1.77824103084698056792029975740, −1.35755729850810324121018470530, −0.981974514750823138388055796368, −0.794593501802062986098040279624, −0.64879909198572218552543301687, 0.64879909198572218552543301687, 0.794593501802062986098040279624, 0.981974514750823138388055796368, 1.35755729850810324121018470530, 1.77824103084698056792029975740, 2.16932305269839800279301302669, 2.22556888988327126909969840085, 2.53774997159346085651828387299, 3.20573841550072706773108418718, 3.25549948398977274777387431850, 3.41509312149737191717507938093, 3.58644321631515063876427990820, 4.27436398000170396485326690680, 4.44014614456768203370353940788, 4.70962905301239260105980502231, 5.14271291967608554379986689233, 5.60505819083254379737755408800, 5.64065692803407489379996253823, 5.69787892197818586327323992626, 5.99189146442465318952986336555, 6.37692104662818563199692830804, 6.54287880014927799329517777402, 6.68396746990042997226212728611, 7.35551899962927700637081966572, 7.67229086395533399586817564586