Dirichlet series
| L(s) = 1 | + 2-s + 3·4-s + 5·5-s + 4·7-s + 2·8-s + 5·10-s − 16·11-s − 8·13-s + 4·14-s + 4·16-s + 17-s − 5·19-s + 15·20-s − 16·22-s − 7·23-s + 5·25-s − 8·26-s + 12·28-s − 5·29-s − 19·31-s + 9·32-s + 34-s + 20·35-s − 37-s − 5·38-s + 10·40-s + 14·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3/2·4-s + 2.23·5-s + 1.51·7-s + 0.707·8-s + 1.58·10-s − 4.82·11-s − 2.21·13-s + 1.06·14-s + 16-s + 0.242·17-s − 1.14·19-s + 3.35·20-s − 3.41·22-s − 1.45·23-s + 25-s − 1.56·26-s + 2.26·28-s − 0.928·29-s − 3.41·31-s + 1.59·32-s + 0.171·34-s + 3.38·35-s − 0.164·37-s − 0.811·38-s + 1.58·40-s + 2.18·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(108.561\) |
| Root analytic conductor: | \(1.34038\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4792058981\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4792058981\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T + 4 p T^{2} - 13 p T^{3} + 31 p T^{4} - 13 p^{2} T^{5} + 4 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( ( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 - 2 T + 12 T^{2} - 15 T^{3} + 61 T^{4} - 15 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( 1 + 16 T + 105 T^{2} + 340 T^{3} + 390 T^{4} - 1332 T^{5} - 9707 T^{6} - 42730 T^{7} - 156045 T^{8} - 42730 p T^{9} - 9707 p^{2} T^{10} - 1332 p^{3} T^{11} + 390 p^{4} T^{12} + 340 p^{5} T^{13} + 105 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 8 T + 17 T^{2} + 14 T^{3} + 276 T^{4} + 578 T^{5} - 4815 T^{6} - 18048 T^{7} - 25061 T^{8} - 18048 p T^{9} - 4815 p^{2} T^{10} + 578 p^{3} T^{11} + 276 p^{4} T^{12} + 14 p^{5} T^{13} + 17 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T - 27 T^{2} + 38 T^{3} + 591 T^{4} - 456 T^{5} - 11720 T^{6} + 3109 T^{7} + 167979 T^{8} + 3109 p T^{9} - 11720 p^{2} T^{10} - 456 p^{3} T^{11} + 591 p^{4} T^{12} + 38 p^{5} T^{13} - 27 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T - 33 T^{2} - 265 T^{3} - 47 T^{4} + 6980 T^{5} + 26814 T^{6} - 68150 T^{7} - 765695 T^{8} - 68150 p T^{9} + 26814 p^{2} T^{10} + 6980 p^{3} T^{11} - 47 p^{4} T^{12} - 265 p^{5} T^{13} - 33 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 7 T + 12 T^{2} + 31 T^{3} + 1026 T^{4} + 672 T^{5} - 34550 T^{6} - 127672 T^{7} - 3411 T^{8} - 127672 p T^{9} - 34550 p^{2} T^{10} + 672 p^{3} T^{11} + 1026 p^{4} T^{12} + 31 p^{5} T^{13} + 12 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T + 12 T^{2} - 110 T^{3} - 322 T^{4} - 4795 T^{5} - 5456 T^{6} - 22500 T^{7} + 487955 T^{8} - 22500 p T^{9} - 5456 p^{2} T^{10} - 4795 p^{3} T^{11} - 322 p^{4} T^{12} - 110 p^{5} T^{13} + 12 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 19 T + 110 T^{2} - 455 T^{3} - 10080 T^{4} - 55148 T^{5} - 24492 T^{6} + 1584870 T^{7} + 12593785 T^{8} + 1584870 p T^{9} - 24492 p^{2} T^{10} - 55148 p^{3} T^{11} - 10080 p^{4} T^{12} - 455 p^{5} T^{13} + 110 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T + 3 T^{2} + 42 T^{3} + 2811 T^{4} + 3486 T^{5} - 59280 T^{6} + 126501 T^{7} + 3465939 T^{8} + 126501 p T^{9} - 59280 p^{2} T^{10} + 3486 p^{3} T^{11} + 2811 p^{4} T^{12} + 42 p^{5} T^{13} + 3 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 7 T + 28 T^{2} - 389 T^{3} + 3975 T^{4} - 389 p T^{5} + 28 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 16 T + 233 T^{2} - 2040 T^{3} + 16241 T^{4} - 2040 p T^{5} + 233 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - T - 62 T^{2} + 78 T^{3} + 3466 T^{4} - 4691 T^{5} - 155900 T^{6} + 95604 T^{7} + 3265179 T^{8} + 95604 p T^{9} - 155900 p^{2} T^{10} - 4691 p^{3} T^{11} + 3466 p^{4} T^{12} + 78 p^{5} T^{13} - 62 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T - 63 T^{2} + 6 T^{3} + 4741 T^{4} + 3522 T^{5} - 240240 T^{6} + 209553 T^{7} + 7555219 T^{8} + 209553 p T^{9} - 240240 p^{2} T^{10} + 3522 p^{3} T^{11} + 4741 p^{4} T^{12} + 6 p^{5} T^{13} - 63 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 30 T + 377 T^{2} + 2475 T^{3} + 8643 T^{4} + 1155 T^{5} - 582521 T^{6} - 10821600 T^{7} - 108687625 T^{8} - 10821600 p T^{9} - 582521 p^{2} T^{10} + 1155 p^{3} T^{11} + 8643 p^{4} T^{12} + 2475 p^{5} T^{13} + 377 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 35 T^{2} + 110 T^{3} + 8070 T^{4} + 33812 T^{5} - 419217 T^{6} - 1871430 T^{7} + 12043555 T^{8} - 1871430 p T^{9} - 419217 p^{2} T^{10} + 33812 p^{3} T^{11} + 8070 p^{4} T^{12} + 110 p^{5} T^{13} + 35 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 4 T - 97 T^{2} + 122 T^{3} + 9681 T^{4} - 8074 T^{5} - 723795 T^{6} + 600576 T^{7} + 39163204 T^{8} + 600576 p T^{9} - 723795 p^{2} T^{10} - 8074 p^{3} T^{11} + 9681 p^{4} T^{12} + 122 p^{5} T^{13} - 97 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 21 T + 305 T^{2} + 2480 T^{3} + 13165 T^{4} - 10682 T^{5} - 810942 T^{6} - 9320005 T^{7} - 80224815 T^{8} - 9320005 p T^{9} - 810942 p^{2} T^{10} - 10682 p^{3} T^{11} + 13165 p^{4} T^{12} + 2480 p^{5} T^{13} + 305 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 2 T - 163 T^{2} - 986 T^{3} + 8481 T^{4} + 176518 T^{5} + 938895 T^{6} - 6925578 T^{7} - 156207836 T^{8} - 6925578 p T^{9} + 938895 p^{2} T^{10} + 176518 p^{3} T^{11} + 8481 p^{4} T^{12} - 986 p^{5} T^{13} - 163 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 30 T + 447 T^{2} + 5690 T^{3} + 63618 T^{4} + 615980 T^{5} + 6355819 T^{6} + 63162750 T^{7} + 565067755 T^{8} + 63162750 p T^{9} + 6355819 p^{2} T^{10} + 615980 p^{3} T^{11} + 63618 p^{4} T^{12} + 5690 p^{5} T^{13} + 447 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 2 T - 123 T^{2} + 626 T^{3} + 1416 T^{4} - 170328 T^{5} + 1146775 T^{6} + 9202888 T^{7} - 131919081 T^{8} + 9202888 p T^{9} + 1146775 p^{2} T^{10} - 170328 p^{3} T^{11} + 1416 p^{4} T^{12} + 626 p^{5} T^{13} - 123 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 77 T^{2} - 700 T^{3} + 12658 T^{4} - 121100 T^{5} + 1037049 T^{6} - 10883950 T^{7} + 150329955 T^{8} - 10883950 p T^{9} + 1037049 p^{2} T^{10} - 121100 p^{3} T^{11} + 12658 p^{4} T^{12} - 700 p^{5} T^{13} + 77 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 + 6 T - 187 T^{2} - 1878 T^{3} + 3816 T^{4} + 286836 T^{5} + 2735335 T^{6} - 15219804 T^{7} - 421030321 T^{8} - 15219804 p T^{9} + 2735335 p^{2} T^{10} + 286836 p^{3} T^{11} + 3816 p^{4} T^{12} - 1878 p^{5} T^{13} - 187 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.52044225796955445810530406344, −5.47802146853100532168494558768, −5.46592068175269420521285820341, −5.43597102201087469605881456947, −5.12728922637482039375752377547, −4.71001723613108896484604163876, −4.52037347494811286217555933965, −4.51096428242402816551181064163, −4.35284859338391175886316586504, −4.30881775836825455913729736707, −4.25892552962089246643049440719, −3.92087209119067758734920433397, −3.28813263893333279505016903947, −3.20078916225827379804735598134, −3.10943904590857521620809716270, −2.90523276492646401083953157883, −2.76180218240884888620676745836, −2.46344087447409336852857776269, −2.36937829162028019823082162308, −2.23824610712082193020406775348, −2.07302494047529127173663374665, −1.88897830038991048164882198480, −1.55833299302875315169646102304, −1.52833873265772016646555835481, −0.15015706164153201670971435032, 0.15015706164153201670971435032, 1.52833873265772016646555835481, 1.55833299302875315169646102304, 1.88897830038991048164882198480, 2.07302494047529127173663374665, 2.23824610712082193020406775348, 2.36937829162028019823082162308, 2.46344087447409336852857776269, 2.76180218240884888620676745836, 2.90523276492646401083953157883, 3.10943904590857521620809716270, 3.20078916225827379804735598134, 3.28813263893333279505016903947, 3.92087209119067758734920433397, 4.25892552962089246643049440719, 4.30881775836825455913729736707, 4.35284859338391175886316586504, 4.51096428242402816551181064163, 4.52037347494811286217555933965, 4.71001723613108896484604163876, 5.12728922637482039375752377547, 5.43597102201087469605881456947, 5.46592068175269420521285820341, 5.47802146853100532168494558768, 5.52044225796955445810530406344
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.