Dirichlet series
| L(s) = 1 | + 3·2-s + 2·3-s + 2·4-s − 4·5-s + 6·6-s + 2·7-s − 3·8-s + 4·9-s − 12·10-s + 4·12-s + 6·14-s − 8·15-s − 3·16-s + 12·17-s + 12·18-s + 9·19-s − 8·20-s + 4·21-s − 27·23-s − 6·24-s + 6·25-s + 4·27-s + 4·28-s − 24·30-s − 21·31-s + 36·34-s − 8·35-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.15·3-s + 4-s − 1.78·5-s + 2.44·6-s + 0.755·7-s − 1.06·8-s + 4/3·9-s − 3.79·10-s + 1.15·12-s + 1.60·14-s − 2.06·15-s − 3/4·16-s + 2.91·17-s + 2.82·18-s + 2.06·19-s − 1.78·20-s + 0.872·21-s − 5.62·23-s − 1.22·24-s + 6/5·25-s + 0.769·27-s + 0.755·28-s − 4.38·30-s − 3.77·31-s + 6.17·34-s − 1.35·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.244191\) |
| Root analytic conductor: | \(0.915657\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.670185990\) |
| \(L(\frac12)\) | \(\approx\) | \(2.670185990\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 2 T + 4 T^{3} - 11 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 + T + T^{2} )^{4} \) | |
| 7 | \( 1 - 2 T + 4 T^{2} + 10 T^{3} - 41 T^{4} + 10 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} - 3 p^{2} T^{5} - p^{3} T^{6} + 9 p^{2} T^{7} - 17 p^{2} T^{8} + 9 p^{3} T^{9} - p^{5} T^{10} - 3 p^{5} T^{11} + p^{8} T^{12} - 3 p^{7} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 1548 p T^{9} + 268 p^{2} T^{10} - 30 p^{3} T^{11} - 2 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 76058 p^{2} T^{10} + 3217 p^{4} T^{12} - 83 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 12 T + 2 p T^{2} - 12 T^{3} + 1078 T^{4} - 6882 T^{5} + 8740 T^{6} - 70272 T^{7} + 637627 T^{8} - 70272 p T^{9} + 8740 p^{2} T^{10} - 6882 p^{3} T^{11} + 1078 p^{4} T^{12} - 12 p^{5} T^{13} + 2 p^{7} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 704196 p T^{9} + 148996 p^{2} T^{10} - 26244 p^{3} T^{11} + 4723 p^{4} T^{12} - 657 p^{5} T^{13} + 100 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 27 T + 17 p T^{2} + 3996 T^{3} + 31651 T^{4} + 205875 T^{5} + 1157938 T^{6} + 5917779 T^{7} + 28782226 T^{8} + 5917779 p T^{9} + 1157938 p^{2} T^{10} + 205875 p^{3} T^{11} + 31651 p^{4} T^{12} + 3996 p^{5} T^{13} + 17 p^{7} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 128951 p^{2} T^{10} + 3250 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 2328618 p T^{9} + 505090 p^{2} T^{10} + 101304 p^{3} T^{11} + 17293 p^{4} T^{12} + 2415 p^{5} T^{13} + 262 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 258098 p T^{9} + 118336 p^{2} T^{10} - 16188 p^{3} T^{11} - 973 p^{4} T^{12} + 493 p^{5} T^{13} - 24 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 1791 p T^{5} + 218 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 1022 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 6 T - 116 T^{2} + 252 T^{3} + 10126 T^{4} - 1986 T^{5} - 595736 T^{6} + 157218 T^{7} + 25623007 T^{8} + 157218 p T^{9} - 595736 p^{2} T^{10} - 1986 p^{3} T^{11} + 10126 p^{4} T^{12} + 252 p^{5} T^{13} - 116 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 24 T + 340 T^{2} + 3552 T^{3} + 29050 T^{4} + 180120 T^{5} + 750160 T^{6} + 1659096 T^{7} + 1273315 T^{8} + 1659096 p T^{9} + 750160 p^{2} T^{10} + 180120 p^{3} T^{11} + 29050 p^{4} T^{12} + 3552 p^{5} T^{13} + 340 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 1772232 p T^{9} - 434420 p^{2} T^{10} - 80082 p^{3} T^{11} + 7690 p^{4} T^{12} + 1164 p^{5} T^{13} - 80 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 5175267 p T^{9} + 816310 p^{2} T^{10} - 141723 p^{3} T^{11} + 19711 p^{4} T^{12} - 2220 p^{5} T^{13} + 223 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 717722 p T^{9} - 3551522 p^{2} T^{10} - 28434 p^{3} T^{11} + 35255 p^{4} T^{12} + 412 p^{5} T^{13} - 234 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 12936548 p^{2} T^{10} + 99532 p^{4} T^{12} - 464 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 - 15 T + 280 T^{2} - 3075 T^{3} + 38779 T^{4} - 422928 T^{5} + 4017052 T^{6} - 39506334 T^{7} + 310273396 T^{8} - 39506334 p T^{9} + 4017052 p^{2} T^{10} - 422928 p^{3} T^{11} + 38779 p^{4} T^{12} - 3075 p^{5} T^{13} + 280 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 29 T + 294 T^{2} + 25 p T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 2403434 p T^{9} + 598654 p^{2} T^{10} + 260496 p^{3} T^{11} + 27377 p^{4} T^{12} + 25 p^{6} T^{13} + 294 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 15 T + 380 T^{2} + 3759 T^{3} + 49260 T^{4} + 3759 p T^{5} + 380 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 26593569 p T^{9} + 3467116 p^{2} T^{10} + 160275 p^{3} T^{11} + 14227 p^{4} T^{12} - 2820 p^{5} T^{13} - 53 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 12257504 p^{2} T^{10} + 81676 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \) | |
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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
−6.39795124932226386872242763150, −6.01726260523854041776089627658, −5.88824414833555773913366801912, −5.77385966398920268550494676436, −5.75186299474872862797220694619, −5.67737766485983331652884373401, −5.35543330983778735793952134542, −5.31195287221177057383474190157, −5.22987993216999564235158100490, −4.79068524653722060066706637218, −4.55965917127511212579555315431, −4.21329370740184668092040061836, −4.14718018282436261902934124379, −4.06771419615551446284927111554, −3.96717157370563445646260609250, −3.79535849299811269075976311713, −3.78056744349844396015475165600, −3.63264221418044970082494965650, −3.16443371488087590418122460373, −2.83839151814065779597054100639, −2.81616806606639162024236732759, −2.31820321714485861841142038213, −2.11159352161850032204025899971, −1.51061224319043289689491793243, −1.24971128301304895122118439524, 1.24971128301304895122118439524, 1.51061224319043289689491793243, 2.11159352161850032204025899971, 2.31820321714485861841142038213, 2.81616806606639162024236732759, 2.83839151814065779597054100639, 3.16443371488087590418122460373, 3.63264221418044970082494965650, 3.78056744349844396015475165600, 3.79535849299811269075976311713, 3.96717157370563445646260609250, 4.06771419615551446284927111554, 4.14718018282436261902934124379, 4.21329370740184668092040061836, 4.55965917127511212579555315431, 4.79068524653722060066706637218, 5.22987993216999564235158100490, 5.31195287221177057383474190157, 5.35543330983778735793952134542, 5.67737766485983331652884373401, 5.75186299474872862797220694619, 5.77385966398920268550494676436, 5.88824414833555773913366801912, 6.01726260523854041776089627658, 6.39795124932226386872242763150
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.