Dirichlet series
| L(s) = 1 | + 8·2-s + 4·3-s + 36·4-s + 2·5-s + 32·6-s − 3·7-s + 120·8-s + 6·9-s + 16·10-s − 6·11-s + 144·12-s − 11·13-s − 24·14-s + 8·15-s + 330·16-s − 8·17-s + 48·18-s + 6·19-s + 72·20-s − 12·21-s − 48·22-s + 20·23-s + 480·24-s + 3·25-s − 88·26-s − 108·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 2.30·3-s + 18·4-s + 0.894·5-s + 13.0·6-s − 1.13·7-s + 42.4·8-s + 2·9-s + 5.05·10-s − 1.80·11-s + 41.5·12-s − 3.05·13-s − 6.41·14-s + 2.06·15-s + 82.5·16-s − 1.94·17-s + 11.3·18-s + 1.37·19-s + 16.0·20-s − 2.61·21-s − 10.2·22-s + 4.17·23-s + 97.9·24-s + 3/5·25-s − 17.2·26-s − 20.4·28-s + 0.371·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(130544.\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(670.4020618\) |
| \(L(\frac12)\) | \(\approx\) | \(670.4020618\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) | |
| 7 | \( 1 + 3 T + 2 T^{2} - 3 p T^{3} - 117 T^{4} - 3 p^{2} T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 + 11 T + 62 T^{2} + 267 T^{3} + 1031 T^{4} + 267 p T^{5} + 62 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 2 T + T^{2} - 6 T^{3} + 12 T^{4} - 12 T^{5} + 159 T^{6} - 16 p^{2} T^{7} + 159 T^{8} - 16 p^{3} T^{9} + 159 p^{2} T^{10} - 12 p^{3} T^{11} + 12 p^{4} T^{12} - 6 p^{5} T^{13} + p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 6 T + 2 T^{2} - 14 T^{3} + 63 T^{4} - 3 p^{2} T^{5} - 2011 T^{6} - 1011 T^{7} + 3345 T^{8} - 1011 p T^{9} - 2011 p^{2} T^{10} - 3 p^{5} T^{11} + 63 p^{4} T^{12} - 14 p^{5} T^{13} + 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 + 4 T + 54 T^{2} + 213 T^{3} + 1257 T^{4} + 213 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( 1 - 6 T - 45 T^{2} + 166 T^{3} + 2262 T^{4} - 236 p T^{5} - 62189 T^{6} + 23010 T^{7} + 1465411 T^{8} + 23010 p T^{9} - 62189 p^{2} T^{10} - 236 p^{4} T^{11} + 2262 p^{4} T^{12} + 166 p^{5} T^{13} - 45 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 10 T + 82 T^{2} - 535 T^{3} + 2547 T^{4} - 535 p T^{5} + 82 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 2 T - 62 T^{2} + 438 T^{3} + 1707 T^{4} - 18759 T^{5} + 38493 T^{6} + 338675 T^{7} - 2208849 T^{8} + 338675 p T^{9} + 38493 p^{2} T^{10} - 18759 p^{3} T^{11} + 1707 p^{4} T^{12} + 438 p^{5} T^{13} - 62 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T - 40 T^{2} + 498 T^{3} - 335 T^{4} - 12051 T^{5} + 37955 T^{6} + 80889 T^{7} - 826703 T^{8} + 80889 p T^{9} + 37955 p^{2} T^{10} - 12051 p^{3} T^{11} - 335 p^{4} T^{12} + 498 p^{5} T^{13} - 40 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 28 T + 408 T^{2} - 3931 T^{3} + 27665 T^{4} - 3931 p T^{5} + 408 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 72 T^{2} + 198 T^{3} + 1259 T^{4} - 11187 T^{5} - 30735 T^{6} + 195921 T^{7} + 3463719 T^{8} + 195921 p T^{9} - 30735 p^{2} T^{10} - 11187 p^{3} T^{11} + 1259 p^{4} T^{12} + 198 p^{5} T^{13} - 72 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 + 6 T - 98 T^{2} - 18 p T^{3} + 5329 T^{4} + 44937 T^{5} - 143171 T^{6} - 901221 T^{7} + 4636105 T^{8} - 901221 p T^{9} - 143171 p^{2} T^{10} + 44937 p^{3} T^{11} + 5329 p^{4} T^{12} - 18 p^{6} T^{13} - 98 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - T - 131 T^{2} - 78 T^{3} + 9356 T^{4} + 11249 T^{5} - 478776 T^{6} - 277502 T^{7} + 21461767 T^{8} - 277502 p T^{9} - 478776 p^{2} T^{10} + 11249 p^{3} T^{11} + 9356 p^{4} T^{12} - 78 p^{5} T^{13} - 131 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 7 T - 119 T^{2} + 1086 T^{3} + 8036 T^{4} - 78157 T^{5} - 258144 T^{6} + 1916824 T^{7} + 10200157 T^{8} + 1916824 p T^{9} - 258144 p^{2} T^{10} - 78157 p^{3} T^{11} + 8036 p^{4} T^{12} + 1086 p^{5} T^{13} - 119 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( ( 1 + 2 T + 93 T^{2} - 306 T^{3} + 4323 T^{4} - 306 p T^{5} + 93 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 - 24 T + 196 T^{2} - 1248 T^{3} + 15194 T^{4} - 68160 T^{5} - 644208 T^{6} + 6345384 T^{7} - 27687421 T^{8} + 6345384 p T^{9} - 644208 p^{2} T^{10} - 68160 p^{3} T^{11} + 15194 p^{4} T^{12} - 1248 p^{5} T^{13} + 196 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 15 T - 72 T^{2} - 1177 T^{3} + 14982 T^{4} + 95414 T^{5} - 1436897 T^{6} - 301836 T^{7} + 148115845 T^{8} - 301836 p T^{9} - 1436897 p^{2} T^{10} + 95414 p^{3} T^{11} + 14982 p^{4} T^{12} - 1177 p^{5} T^{13} - 72 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 6 T - 153 T^{2} + 198 T^{3} + 15790 T^{4} + 18060 T^{5} - 946377 T^{6} - 602562 T^{7} + 42720099 T^{8} - 602562 p T^{9} - 946377 p^{2} T^{10} + 18060 p^{3} T^{11} + 15790 p^{4} T^{12} + 198 p^{5} T^{13} - 153 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - T - 284 T^{2} + 143 T^{3} + 50058 T^{4} - 14706 T^{5} - 5753901 T^{6} + 372816 T^{7} + 494276869 T^{8} + 372816 p T^{9} - 5753901 p^{2} T^{10} - 14706 p^{3} T^{11} + 50058 p^{4} T^{12} + 143 p^{5} T^{13} - 284 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 12 T - 95 T^{2} - 1656 T^{3} + 26 p T^{4} + 49164 T^{5} - 718131 T^{6} + 1559838 T^{7} + 122032043 T^{8} + 1559838 p T^{9} - 718131 p^{2} T^{10} + 49164 p^{3} T^{11} + 26 p^{5} T^{12} - 1656 p^{5} T^{13} - 95 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 16 T + 246 T^{2} + 2655 T^{3} + 28977 T^{4} + 2655 p T^{5} + 246 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( ( 1 + 25 T + 526 T^{2} + 6855 T^{3} + 77813 T^{4} + 6855 p T^{5} + 526 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + T - 199 T^{2} - 634 T^{3} + 12828 T^{4} + 70339 T^{5} - 1453960 T^{6} - 2228292 T^{7} + 235898857 T^{8} - 2228292 p T^{9} - 1453960 p^{2} T^{10} + 70339 p^{3} T^{11} + 12828 p^{4} T^{12} - 634 p^{5} T^{13} - 199 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.79398177847373631261426592140, −4.50917239324284242106705905436, −4.32745435039998707366762329490, −4.31328749991812180600579542339, −4.27595616495077692720184657183, −4.10909715035425819767296493703, −3.96952982485927793554815017021, −3.90404185854603470782363961804, −3.69503428115492167533167241869, −3.14721144183630960173121111191, −3.04025863020171689780142006450, −3.02494871357376634597234412654, −2.97650281298064673523593771498, −2.96011895992169753206188018258, −2.78642900368569589272471506941, −2.62906258487138382258670146514, −2.52873782507111515886733830617, −2.50219283918103932188771879971, −2.45696151963468978573467466366, −2.11600551289411832966245222616, −2.09574131163699374491228031648, −1.52684578173480360360349497596, −1.22026042075076759750175896581, −1.16357820883724181801072898118, −0.72278464043229839220151938478, 0.72278464043229839220151938478, 1.16357820883724181801072898118, 1.22026042075076759750175896581, 1.52684578173480360360349497596, 2.09574131163699374491228031648, 2.11600551289411832966245222616, 2.45696151963468978573467466366, 2.50219283918103932188771879971, 2.52873782507111515886733830617, 2.62906258487138382258670146514, 2.78642900368569589272471506941, 2.96011895992169753206188018258, 2.97650281298064673523593771498, 3.02494871357376634597234412654, 3.04025863020171689780142006450, 3.14721144183630960173121111191, 3.69503428115492167533167241869, 3.90404185854603470782363961804, 3.96952982485927793554815017021, 4.10909715035425819767296493703, 4.27595616495077692720184657183, 4.31328749991812180600579542339, 4.32745435039998707366762329490, 4.50917239324284242106705905436, 4.79398177847373631261426592140
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.