Dirichlet series
| L(s) = 1 | + 4·2-s + 7·4-s + 2·5-s − 3·7-s + 8·8-s + 8·10-s + 5·11-s + 4·13-s − 12·14-s + 4·16-s − 17-s − 19-s + 14·20-s + 20·22-s + 18·23-s + 25-s + 16·26-s − 21·28-s − 19·29-s + 6·31-s − 4·32-s − 4·34-s − 6·35-s + 4·37-s − 4·38-s + 16·40-s + 4·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 7/2·4-s + 0.894·5-s − 1.13·7-s + 2.82·8-s + 2.52·10-s + 1.50·11-s + 1.10·13-s − 3.20·14-s + 16-s − 0.242·17-s − 0.229·19-s + 3.13·20-s + 4.26·22-s + 3.75·23-s + 1/5·25-s + 3.13·26-s − 3.96·28-s − 3.52·29-s + 1.07·31-s − 0.707·32-s − 0.685·34-s − 1.01·35-s + 0.657·37-s − 0.648·38-s + 2.52·40-s + 0.624·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(59574.2\) |
| Root analytic conductor: | \(1.98811\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(33.22870216\) |
| \(L(\frac12)\) | \(\approx\) | \(33.22870216\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 5 T + 26 T^{2} - 45 T^{3} + 191 T^{4} - 45 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 - p^{2} T + 9 T^{2} - p^{4} T^{3} + 29 T^{4} - 7 p^{3} T^{5} + 93 T^{6} - p^{7} T^{7} + 173 T^{8} - p^{8} T^{9} + 93 p^{2} T^{10} - 7 p^{6} T^{11} + 29 p^{4} T^{12} - p^{9} T^{13} + 9 p^{6} T^{14} - p^{9} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 3 T + 5 T^{2} - 11 T^{3} - 29 T^{4} + 62 T^{5} + 96 T^{6} - 690 T^{7} - 5491 T^{8} - 690 p T^{9} + 96 p^{2} T^{10} + 62 p^{3} T^{11} - 29 p^{4} T^{12} - 11 p^{5} T^{13} + 5 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 7 T^{2} - 59 T^{3} + 437 T^{4} - 1241 T^{5} + 2351 T^{6} - 18028 T^{7} + 93877 T^{8} - 18028 p T^{9} + 2351 p^{2} T^{10} - 1241 p^{3} T^{11} + 437 p^{4} T^{12} - 59 p^{5} T^{13} + 7 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + T - 36 T^{2} + 29 T^{3} + 574 T^{4} - 716 T^{5} + 2868 T^{6} + 666 p T^{7} - 138837 T^{8} + 666 p^{2} T^{9} + 2868 p^{2} T^{10} - 716 p^{3} T^{11} + 574 p^{4} T^{12} + 29 p^{5} T^{13} - 36 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + T - 7 T^{2} - 96 T^{3} + 293 T^{4} - 994 T^{5} - 1110 T^{6} - 17535 T^{7} + 246561 T^{8} - 17535 p T^{9} - 1110 p^{2} T^{10} - 994 p^{3} T^{11} + 293 p^{4} T^{12} - 96 p^{5} T^{13} - 7 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 9 T + 38 T^{2} + 85 T^{3} - 979 T^{4} + 85 p T^{5} + 38 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 19 T + 113 T^{2} + 86 T^{3} - 1117 T^{4} - 5456 T^{5} - 33160 T^{6} + 205495 T^{7} + 3154571 T^{8} + 205495 p T^{9} - 33160 p^{2} T^{10} - 5456 p^{3} T^{11} - 1117 p^{4} T^{12} + 86 p^{5} T^{13} + 113 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T - 5 T^{2} + 15 T^{3} + 1295 T^{4} + 357 T^{5} - 47197 T^{6} + 151560 T^{7} + 294595 T^{8} + 151560 p T^{9} - 47197 p^{2} T^{10} + 357 p^{3} T^{11} + 1295 p^{4} T^{12} + 15 p^{5} T^{13} - 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 4 T - 97 T^{2} + 462 T^{3} + 2066 T^{4} - 23894 T^{5} + 174045 T^{6} + 437226 T^{7} - 12204441 T^{8} + 437226 p T^{9} + 174045 p^{2} T^{10} - 23894 p^{3} T^{11} + 2066 p^{4} T^{12} + 462 p^{5} T^{13} - 97 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4 T - 85 T^{2} - 20 T^{3} + 2540 T^{4} + 13928 T^{5} + 163313 T^{6} - 521020 T^{7} - 12527665 T^{8} - 521020 p T^{9} + 163313 p^{2} T^{10} + 13928 p^{3} T^{11} + 2540 p^{4} T^{12} - 20 p^{5} T^{13} - 85 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 21 T + 293 T^{2} - 2900 T^{3} + 21441 T^{4} - 2900 p T^{5} + 293 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 4 T - 7 T^{2} + 23 T^{3} + 3951 T^{4} + 7279 T^{5} - 163935 T^{6} - 137206 T^{7} + 7103639 T^{8} - 137206 p T^{9} - 163935 p^{2} T^{10} + 7279 p^{3} T^{11} + 3951 p^{4} T^{12} + 23 p^{5} T^{13} - 7 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 3 T - 69 T^{2} - 487 T^{3} + 4629 T^{4} + 2812 T^{5} - 362068 T^{6} + 185784 T^{7} + 29291723 T^{8} + 185784 p T^{9} - 362068 p^{2} T^{10} + 2812 p^{3} T^{11} + 4629 p^{4} T^{12} - 487 p^{5} T^{13} - 69 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 19 T + 83 T^{2} - 161 T^{3} + 7463 T^{4} - 113974 T^{5} + 1180460 T^{6} - 5430220 T^{7} + 5701721 T^{8} - 5430220 p T^{9} + 1180460 p^{2} T^{10} - 113974 p^{3} T^{11} + 7463 p^{4} T^{12} - 161 p^{5} T^{13} + 83 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 2 T - 48 T^{2} - 78 T^{3} + 7033 T^{4} + 9842 T^{5} - 572520 T^{6} + 148460 T^{7} + 27924921 T^{8} + 148460 p T^{9} - 572520 p^{2} T^{10} + 9842 p^{3} T^{11} + 7033 p^{4} T^{12} - 78 p^{5} T^{13} - 48 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 40 T + 727 T^{2} + 8050 T^{3} + 56358 T^{4} + 99810 T^{5} - 3707511 T^{6} - 61987150 T^{7} - 612781425 T^{8} - 61987150 p T^{9} - 3707511 p^{2} T^{10} + 99810 p^{3} T^{11} + 56358 p^{4} T^{12} + 8050 p^{5} T^{13} + 727 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 23 T + 222 T^{2} + 2419 T^{3} + 36136 T^{4} + 379918 T^{5} + 3242270 T^{6} + 31455482 T^{7} + 296606219 T^{8} + 31455482 p T^{9} + 3242270 p^{2} T^{10} + 379918 p^{3} T^{11} + 36136 p^{4} T^{12} + 2419 p^{5} T^{13} + 222 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 17 T - 4 T^{2} + 386 T^{3} + 11750 T^{4} - 154077 T^{5} + 1787170 T^{6} - 7920480 T^{7} - 39647009 T^{8} - 7920480 p T^{9} + 1787170 p^{2} T^{10} - 154077 p^{3} T^{11} + 11750 p^{4} T^{12} + 386 p^{5} T^{13} - 4 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 25 T + 105 T^{2} + 2035 T^{3} - 19109 T^{4} + 48470 T^{5} - 251250 T^{6} - 10913400 T^{7} + 221937861 T^{8} - 10913400 p T^{9} - 251250 p^{2} T^{10} + 48470 p^{3} T^{11} - 19109 p^{4} T^{12} + 2035 p^{5} T^{13} + 105 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 206 T^{2} + 400 T^{3} + 21551 T^{4} + 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 12 T - 15 T^{2} - 156 T^{3} + 22446 T^{4} - 71028 T^{5} - 2258399 T^{6} + 7047930 T^{7} + 179384859 T^{8} + 7047930 p T^{9} - 2258399 p^{2} T^{10} - 71028 p^{3} T^{11} + 22446 p^{4} T^{12} - 156 p^{5} T^{13} - 15 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.69699435305986426476824678844, −4.69418115073864446081323110932, −4.56916356801682272099551608744, −4.53100671009751216785595535850, −4.29773418164837940494407702290, −4.05904487407791995470187751280, −3.88909692121559615266172526868, −3.79966107083147071871744925328, −3.74426920170486163352982701414, −3.69294958205671947613563480489, −3.48923574351933473897084616534, −3.36872545623079592588262841395, −3.19200570804083067385546548662, −2.64916746568538197057958707652, −2.63133917044407425843298711651, −2.59781971091851924441181336374, −2.55480531449590777837926476546, −2.38197460642877296967112792569, −2.36952804555738933025342172624, −1.58366093775201601225502476770, −1.48667697873182869157171774834, −1.39597753843246940710286979039, −1.29692810728268680354357732756, −0.69942163222639236284108520211, −0.62739843019553103253884322924, 0.62739843019553103253884322924, 0.69942163222639236284108520211, 1.29692810728268680354357732756, 1.39597753843246940710286979039, 1.48667697873182869157171774834, 1.58366093775201601225502476770, 2.36952804555738933025342172624, 2.38197460642877296967112792569, 2.55480531449590777837926476546, 2.59781971091851924441181336374, 2.63133917044407425843298711651, 2.64916746568538197057958707652, 3.19200570804083067385546548662, 3.36872545623079592588262841395, 3.48923574351933473897084616534, 3.69294958205671947613563480489, 3.74426920170486163352982701414, 3.79966107083147071871744925328, 3.88909692121559615266172526868, 4.05904487407791995470187751280, 4.29773418164837940494407702290, 4.53100671009751216785595535850, 4.56916356801682272099551608744, 4.69418115073864446081323110932, 4.69699435305986426476824678844
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.