Dirichlet series
| L(s) = 1 | − 4·2-s − 8·3-s + 6·4-s + 2·5-s + 32·6-s + 3·7-s + 36·9-s − 8·10-s + 12·11-s − 48·12-s − 11·13-s − 12·14-s − 16·15-s − 15·16-s + 4·17-s − 144·18-s − 12·19-s + 12·20-s − 24·21-s − 48·22-s − 10·23-s + 3·25-s + 44·26-s − 120·27-s + 18·28-s + 2·29-s + 64·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 4.61·3-s + 3·4-s + 0.894·5-s + 13.0·6-s + 1.13·7-s + 12·9-s − 2.52·10-s + 3.61·11-s − 13.8·12-s − 3.05·13-s − 3.20·14-s − 4.13·15-s − 3.75·16-s + 0.970·17-s − 33.9·18-s − 2.75·19-s + 2.68·20-s − 5.23·21-s − 10.2·22-s − 2.08·23-s + 3/5·25-s + 8.62·26-s − 23.0·27-s + 3.40·28-s + 0.371·29-s + 11.6·30-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
| \(L(1)\) | \(\approx\) | \(0.04156501096\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04156501096\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 7 | \( 1 - 3 T + 8 T^{2} - 33 T^{3} + 123 T^{4} - 33 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 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 | \( ( 1 - 6 T + 34 T^{2} - 109 T^{3} + 439 T^{4} - 109 p T^{5} + 34 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 - 4 T - 38 T^{2} + 210 T^{3} + 807 T^{4} - 5067 T^{5} - 5781 T^{6} + 41251 T^{7} + 39747 T^{8} + 41251 p T^{9} - 5781 p^{2} T^{10} - 5067 p^{3} T^{11} + 807 p^{4} T^{12} + 210 p^{5} T^{13} - 38 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + 6 T + 81 T^{2} + 326 T^{3} + 2343 T^{4} + 326 p T^{5} + 81 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 + 10 T + 18 T^{2} - 250 T^{3} - 51 p T^{4} + 5235 T^{5} + 41077 T^{6} - 99925 T^{7} - 1499403 T^{8} - 99925 p T^{9} + 41077 p^{2} T^{10} + 5235 p^{3} T^{11} - 51 p^{5} T^{12} - 250 p^{5} T^{13} + 18 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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 | \( 1 + 28 T + 376 T^{2} + 3562 T^{3} + 28731 T^{4} + 200055 T^{5} + 1210029 T^{6} + 7124103 T^{7} + 43260595 T^{8} + 7124103 p T^{9} + 1210029 p^{2} T^{10} + 200055 p^{3} T^{11} + 28731 p^{4} T^{12} + 3562 p^{5} T^{13} + 376 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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 | \( 1 - 2 T - 89 T^{2} - 798 T^{3} + 4938 T^{4} + 63804 T^{5} + 375171 T^{6} - 3093430 T^{7} - 27417693 T^{8} - 3093430 p T^{9} + 375171 p^{2} T^{10} + 63804 p^{3} T^{11} + 4938 p^{4} T^{12} - 798 p^{5} T^{13} - 89 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 24 T + 380 T^{2} + 3936 T^{3} + 34742 T^{4} + 3936 p T^{5} + 380 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 15 T + 297 T^{2} - 2816 T^{3} + 30987 T^{4} - 2816 p T^{5} + 297 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 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 | \( 1 - 25 T + 99 T^{2} + 560 T^{3} + 27488 T^{4} - 325175 T^{5} - 858096 T^{6} - 13373550 T^{7} + 499580157 T^{8} - 13373550 p T^{9} - 858096 p^{2} T^{10} - 325175 p^{3} T^{11} + 27488 p^{4} T^{12} + 560 p^{5} T^{13} + 99 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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
−4.79903305792873833433470077575, −4.60459595486079229132379207505, −4.58728335245424299646857759940, −4.56928656113881276190019579225, −4.29881591152089533692565540595, −4.17979761328932001876527162970, −4.13518349786327734910885763731, −3.79870671876247846421497330350, −3.73882189180462111423058391629, −3.56167057514259213955537003699, −3.42179694352292648995554408399, −3.13145363097081387844168769878, −2.91632931095892330012021134408, −2.37038496840894732881672125353, −2.26751345546645888168391719033, −2.23961935161897979127902966087, −1.77220443847729937085148007749, −1.71973548311791833033901727978, −1.71097432284838726656442326217, −1.51854599648780713647240269983, −1.37679228023179005986969849798, −1.06546948865564173084637249708, −0.67215294790474298877177235815, −0.38157355843925044350891784353, −0.23412746787435347712020068497, 0.23412746787435347712020068497, 0.38157355843925044350891784353, 0.67215294790474298877177235815, 1.06546948865564173084637249708, 1.37679228023179005986969849798, 1.51854599648780713647240269983, 1.71097432284838726656442326217, 1.71973548311791833033901727978, 1.77220443847729937085148007749, 2.23961935161897979127902966087, 2.26751345546645888168391719033, 2.37038496840894732881672125353, 2.91632931095892330012021134408, 3.13145363097081387844168769878, 3.42179694352292648995554408399, 3.56167057514259213955537003699, 3.73882189180462111423058391629, 3.79870671876247846421497330350, 4.13518349786327734910885763731, 4.17979761328932001876527162970, 4.29881591152089533692565540595, 4.56928656113881276190019579225, 4.58728335245424299646857759940, 4.60459595486079229132379207505, 4.79903305792873833433470077575
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.