Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·5-s − 4·6-s − 2·7-s + 9-s − 8·10-s − 5·11-s − 2·12-s + 6·13-s − 4·14-s + 8·15-s − 2·17-s + 2·18-s + 8·19-s − 4·20-s + 4·21-s − 10·22-s − 20·23-s + 15·25-s + 12·26-s − 2·28-s − 18·29-s + 16·30-s + 9·31-s − 2·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s − 1.63·6-s − 0.755·7-s + 1/3·9-s − 2.52·10-s − 1.50·11-s − 0.577·12-s + 1.66·13-s − 1.06·14-s + 2.06·15-s − 0.485·17-s + 0.471·18-s + 1.83·19-s − 0.894·20-s + 0.872·21-s − 2.13·22-s − 4.17·23-s + 3·25-s + 2.35·26-s − 0.377·28-s − 3.34·29-s + 2.92·30-s + 1.61·31-s − 0.353·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.23591\) |
| Root analytic conductor: | \(1.09442\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06871855766\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06871855766\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 5 | \( 1 + 4 T + T^{2} - 16 T^{3} - 39 T^{4} - 16 p T^{5} + p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( ( 1 + T + 9 T^{2} + 17 T^{3} + 104 T^{4} + 17 p T^{5} + 9 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 5 T + 38 T^{2} + 210 T^{3} + 1048 T^{4} + 405 p T^{5} + 19301 T^{6} + 69050 T^{7} + 243280 T^{8} + 69050 p T^{9} + 19301 p^{2} T^{10} + 405 p^{4} T^{11} + 1048 p^{4} T^{12} + 210 p^{5} T^{13} + 38 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 6 T + 6 T^{2} + 2 p T^{3} + 3 T^{4} - 1366 T^{5} + 5284 T^{6} - 2112 T^{7} - 15227 T^{8} - 2112 p T^{9} + 5284 p^{2} T^{10} - 1366 p^{3} T^{11} + 3 p^{4} T^{12} + 2 p^{6} T^{13} + 6 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T + 44 T^{2} + 12 T^{3} + 1273 T^{4} - 258 T^{5} + 28916 T^{6} - 13984 T^{7} + 544693 T^{8} - 13984 p T^{9} + 28916 p^{2} T^{10} - 258 p^{3} T^{11} + 1273 p^{4} T^{12} + 12 p^{5} T^{13} + 44 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 62 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( ( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 200 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 18 T + 160 T^{2} + 1050 T^{3} + 5925 T^{4} + 26514 T^{5} + 92732 T^{6} + 300180 T^{7} + 1280525 T^{8} + 300180 p T^{9} + 92732 p^{2} T^{10} + 26514 p^{3} T^{11} + 5925 p^{4} T^{12} + 1050 p^{5} T^{13} + 160 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 9 T - 20 T^{2} + 300 T^{3} + 690 T^{4} - 8217 T^{5} - 2797 T^{6} + 139770 T^{7} - 752500 T^{8} + 139770 p T^{9} - 2797 p^{2} T^{10} - 8217 p^{3} T^{11} + 690 p^{4} T^{12} + 300 p^{5} T^{13} - 20 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 21 T + 258 T^{2} - 2462 T^{3} + 19746 T^{4} - 143741 T^{5} + 980785 T^{6} - 6308436 T^{7} + 38864404 T^{8} - 6308436 p T^{9} + 980785 p^{2} T^{10} - 143741 p^{3} T^{11} + 19746 p^{4} T^{12} - 2462 p^{5} T^{13} + 258 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 2 T + 36 T^{2} - 168 T^{3} + 2201 T^{4} - 17286 T^{5} + 84044 T^{6} - 959864 T^{7} + 2420157 T^{8} - 959864 p T^{9} + 84044 p^{2} T^{10} - 17286 p^{3} T^{11} + 2201 p^{4} T^{12} - 168 p^{5} T^{13} + 36 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 16 T + 168 T^{2} + 1280 T^{3} + 9326 T^{4} + 1280 p T^{5} + 168 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 10 T + 66 T^{2} + 210 T^{3} + 1787 T^{4} - 6510 T^{5} - 94072 T^{6} - 974600 T^{7} - 2764935 T^{8} - 974600 p T^{9} - 94072 p^{2} T^{10} - 6510 p^{3} T^{11} + 1787 p^{4} T^{12} + 210 p^{5} T^{13} + 66 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 7 T + 62 T^{2} - 816 T^{3} + 3586 T^{4} + 23943 T^{5} - 77365 T^{6} + 2388872 T^{7} - 30975296 T^{8} + 2388872 p T^{9} - 77365 p^{2} T^{10} + 23943 p^{3} T^{11} + 3586 p^{4} T^{12} - 816 p^{5} T^{13} + 62 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 25 T + 212 T^{2} + 150 T^{3} - 12692 T^{4} - 128475 T^{5} - 252121 T^{6} + 7858000 T^{7} + 98067400 T^{8} + 7858000 p T^{9} - 252121 p^{2} T^{10} - 128475 p^{3} T^{11} - 12692 p^{4} T^{12} + 150 p^{5} T^{13} + 212 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 10 T - 62 T^{2} + 700 T^{3} + 5543 T^{4} - 26410 T^{5} - 544444 T^{6} + 851800 T^{7} + 32679625 T^{8} + 851800 p T^{9} - 544444 p^{2} T^{10} - 26410 p^{3} T^{11} + 5543 p^{4} T^{12} + 700 p^{5} T^{13} - 62 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 2 T + 34 T^{2} + 232 T^{3} - 637 T^{4} - 10798 T^{5} + 66956 T^{6} + 2488096 T^{7} - 7257587 T^{8} + 2488096 p T^{9} + 66956 p^{2} T^{10} - 10798 p^{3} T^{11} - 637 p^{4} T^{12} + 232 p^{5} T^{13} + 34 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 182 T^{2} - 750 T^{3} + 7983 T^{4} + 134400 T^{5} + 952256 T^{6} - 5830800 T^{7} - 134438695 T^{8} - 5830800 p T^{9} + 952256 p^{2} T^{10} + 134400 p^{3} T^{11} + 7983 p^{4} T^{12} - 750 p^{5} T^{13} - 182 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 + 24 T + 226 T^{2} + 1116 T^{3} + 5643 T^{4} + 16584 T^{5} - 622036 T^{6} - 12920832 T^{7} - 137738827 T^{8} - 12920832 p T^{9} - 622036 p^{2} T^{10} + 16584 p^{3} T^{11} + 5643 p^{4} T^{12} + 1116 p^{5} T^{13} + 226 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 6 T + 34 T^{2} + 336 T^{3} + 4761 T^{4} - 34122 T^{5} + 60506 T^{6} - 838512 T^{7} + 1917227 T^{8} - 838512 p T^{9} + 60506 p^{2} T^{10} - 34122 p^{3} T^{11} + 4761 p^{4} T^{12} + 336 p^{5} T^{13} + 34 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 11 T - 4 T^{2} - 264 T^{3} + 5188 T^{4} + 21759 T^{5} + 513329 T^{6} - 7685372 T^{7} + 27478708 T^{8} - 7685372 p T^{9} + 513329 p^{2} T^{10} + 21759 p^{3} T^{11} + 5188 p^{4} T^{12} - 264 p^{5} T^{13} - 4 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 9 T - 166 T^{2} + 1356 T^{3} + 12786 T^{4} - 119727 T^{5} + 426031 T^{6} + 2824668 T^{7} - 98110168 T^{8} + 2824668 p T^{9} + 426031 p^{2} T^{10} - 119727 p^{3} T^{11} + 12786 p^{4} T^{12} + 1356 p^{5} T^{13} - 166 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - T + 28 T^{2} - 632 T^{3} + 5996 T^{4} - 78691 T^{5} + 707075 T^{6} - 3586376 T^{7} + 72919024 T^{8} - 3586376 p T^{9} + 707075 p^{2} T^{10} - 78691 p^{3} T^{11} + 5996 p^{4} T^{12} - 632 p^{5} T^{13} + 28 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.16203384382464938825589329441, −5.81986639519649156966693179357, −5.77741574399437854828444367026, −5.45710638119489757845118112524, −5.21539843795396711299561405576, −5.18756501823203964174365978041, −5.09864949159950742781207146426, −4.90525686306065385207265206139, −4.80276738030628124787787368750, −4.35867395292842845993495595748, −4.22323358205086127576466658121, −4.17813204650347352067016675566, −4.14198077624082652590633160123, −3.91396595650554236474837334782, −3.61697489728026623028736186962, −3.32682123668003054015689684564, −3.21643557336723267983573575044, −3.17168820940116161791242581814, −2.95160996790377939743423264063, −2.75874616185726000106929453455, −2.08959845916282485844296962097, −1.99240304180795183403286585906, −1.63895400169781869821855340911, −1.22786740220948346480124704589, −0.12975565448505981835450572316, 0.12975565448505981835450572316, 1.22786740220948346480124704589, 1.63895400169781869821855340911, 1.99240304180795183403286585906, 2.08959845916282485844296962097, 2.75874616185726000106929453455, 2.95160996790377939743423264063, 3.17168820940116161791242581814, 3.21643557336723267983573575044, 3.32682123668003054015689684564, 3.61697489728026623028736186962, 3.91396595650554236474837334782, 4.14198077624082652590633160123, 4.17813204650347352067016675566, 4.22323358205086127576466658121, 4.35867395292842845993495595748, 4.80276738030628124787787368750, 4.90525686306065385207265206139, 5.09864949159950742781207146426, 5.18756501823203964174365978041, 5.21539843795396711299561405576, 5.45710638119489757845118112524, 5.77741574399437854828444367026, 5.81986639519649156966693179357, 6.16203384382464938825589329441
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.