Dirichlet series
| L(s) = 1 | + 8·2-s − 8·3-s + 36·4-s + 5-s − 64·6-s + 120·8-s + 36·9-s + 8·10-s + 3·11-s − 288·12-s + 9·13-s − 8·15-s + 330·16-s + 8·17-s + 288·18-s + 36·20-s + 24·22-s + 8·23-s − 960·24-s − 13·25-s + 72·26-s − 120·27-s + 8·29-s − 64·30-s − 9·31-s + 792·32-s − 24·33-s + ⋯ |
| L(s) = 1 | + 5.65·2-s − 4.61·3-s + 18·4-s + 0.447·5-s − 26.1·6-s + 42.4·8-s + 12·9-s + 2.52·10-s + 0.904·11-s − 83.1·12-s + 2.49·13-s − 2.06·15-s + 82.5·16-s + 1.94·17-s + 67.8·18-s + 8.04·20-s + 5.11·22-s + 1.66·23-s − 195.·24-s − 2.59·25-s + 14.1·26-s − 23.0·27-s + 1.48·29-s − 11.6·30-s − 1.61·31-s + 140.·32-s − 4.17·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.08751\times 10^{12}\) |
| Root analytic conductor: | \(5.65297\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 23^{8} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(953.3263596\) |
| \(L(\frac12)\) | \(\approx\) | \(953.3263596\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 23 | \( ( 1 - T )^{8} \) | |
| 29 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 - T + 14 T^{2} - 31 T^{3} + 129 T^{4} - 312 T^{5} + 994 T^{6} - 408 p T^{7} + 5724 T^{8} - 408 p^{2} T^{9} + 994 p^{2} T^{10} - 312 p^{3} T^{11} + 129 p^{4} T^{12} - 31 p^{5} T^{13} + 14 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 25 T^{2} + 10 T^{3} + 326 T^{4} + 26 p T^{5} + 3343 T^{6} + 1532 T^{7} + 27186 T^{8} + 1532 p T^{9} + 3343 p^{2} T^{10} + 26 p^{4} T^{11} + 326 p^{4} T^{12} + 10 p^{5} T^{13} + 25 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 - 3 T + 29 T^{2} + 2 T^{3} + 382 T^{4} + 258 T^{5} + 7307 T^{6} + 1497 T^{7} + 90306 T^{8} + 1497 p T^{9} + 7307 p^{2} T^{10} + 258 p^{3} T^{11} + 382 p^{4} T^{12} + 2 p^{5} T^{13} + 29 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 9 T + 81 T^{2} - 452 T^{3} + 2668 T^{4} - 11664 T^{5} + 54123 T^{6} - 199599 T^{7} + 800542 T^{8} - 199599 p T^{9} + 54123 p^{2} T^{10} - 11664 p^{3} T^{11} + 2668 p^{4} T^{12} - 452 p^{5} T^{13} + 81 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 8 T + 97 T^{2} - 638 T^{3} + 4566 T^{4} - 24906 T^{5} + 135183 T^{6} - 619840 T^{7} + 2744882 T^{8} - 619840 p T^{9} + 135183 p^{2} T^{10} - 24906 p^{3} T^{11} + 4566 p^{4} T^{12} - 638 p^{5} T^{13} + 97 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 121 T^{2} + 10 T^{3} + 6830 T^{4} + 782 T^{5} + 235447 T^{6} + 26684 T^{7} + 5414946 T^{8} + 26684 p T^{9} + 235447 p^{2} T^{10} + 782 p^{3} T^{11} + 6830 p^{4} T^{12} + 10 p^{5} T^{13} + 121 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 + 9 T + 79 T^{2} + 382 T^{3} + 1446 T^{4} + 4854 T^{5} + 9449 T^{6} - 76151 T^{7} - 291102 T^{8} - 76151 p T^{9} + 9449 p^{2} T^{10} + 4854 p^{3} T^{11} + 1446 p^{4} T^{12} + 382 p^{5} T^{13} + 79 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 7 T + 128 T^{2} - 541 T^{3} + 7041 T^{4} - 19856 T^{5} + 287290 T^{6} - 682032 T^{7} + 10991528 T^{8} - 682032 p T^{9} + 287290 p^{2} T^{10} - 19856 p^{3} T^{11} + 7041 p^{4} T^{12} - 541 p^{5} T^{13} + 128 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 3 T + 178 T^{2} - 85 T^{3} + 12967 T^{4} + 38400 T^{5} + 522424 T^{6} + 4049448 T^{7} + 18123068 T^{8} + 4049448 p T^{9} + 522424 p^{2} T^{10} + 38400 p^{3} T^{11} + 12967 p^{4} T^{12} - 85 p^{5} T^{13} + 178 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 16 T + 269 T^{2} - 2994 T^{3} + 34966 T^{4} - 306106 T^{5} + 2700755 T^{6} - 19504080 T^{7} + 141734978 T^{8} - 19504080 p T^{9} + 2700755 p^{2} T^{10} - 306106 p^{3} T^{11} + 34966 p^{4} T^{12} - 2994 p^{5} T^{13} + 269 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 24 T + 449 T^{2} - 6138 T^{3} + 72934 T^{4} - 736450 T^{5} + 6670735 T^{6} - 53290128 T^{7} + 387139666 T^{8} - 53290128 p T^{9} + 6670735 p^{2} T^{10} - 736450 p^{3} T^{11} + 72934 p^{4} T^{12} - 6138 p^{5} T^{13} + 449 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T + 292 T^{2} - 1780 T^{3} + 40676 T^{4} - 209220 T^{5} + 3660732 T^{6} - 16048752 T^{7} + 229566358 T^{8} - 16048752 p T^{9} + 3660732 p^{2} T^{10} - 209220 p^{3} T^{11} + 40676 p^{4} T^{12} - 1780 p^{5} T^{13} + 292 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 3 T + 150 T^{2} + 349 T^{3} + 13673 T^{4} + 12544 T^{5} + 816478 T^{6} - 190746 T^{7} + 48114580 T^{8} - 190746 p T^{9} + 816478 p^{2} T^{10} + 12544 p^{3} T^{11} + 13673 p^{4} T^{12} + 349 p^{5} T^{13} + 150 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 31 T + 643 T^{2} - 9466 T^{3} + 118324 T^{4} - 1268360 T^{5} + 12490781 T^{6} - 110698579 T^{7} + 910240006 T^{8} - 110698579 p T^{9} + 12490781 p^{2} T^{10} - 1268360 p^{3} T^{11} + 118324 p^{4} T^{12} - 9466 p^{5} T^{13} + 643 p^{6} T^{14} - 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 11 T + 233 T^{2} + 1726 T^{3} + 28104 T^{4} + 179316 T^{5} + 2485199 T^{6} + 13095181 T^{7} + 173475598 T^{8} + 13095181 p T^{9} + 2485199 p^{2} T^{10} + 179316 p^{3} T^{11} + 28104 p^{4} T^{12} + 1726 p^{5} T^{13} + 233 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 7 T + 399 T^{2} - 2354 T^{3} + 77214 T^{4} - 388770 T^{5} + 9468465 T^{6} - 40765031 T^{7} + 800730306 T^{8} - 40765031 p T^{9} + 9468465 p^{2} T^{10} - 388770 p^{3} T^{11} + 77214 p^{4} T^{12} - 2354 p^{5} T^{13} + 399 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 14 T + 240 T^{2} - 2054 T^{3} + 23828 T^{4} - 200586 T^{5} + 2457648 T^{6} - 19847266 T^{7} + 214335958 T^{8} - 19847266 p T^{9} + 2457648 p^{2} T^{10} - 200586 p^{3} T^{11} + 23828 p^{4} T^{12} - 2054 p^{5} T^{13} + 240 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 12 T + 282 T^{2} - 1812 T^{3} + 29608 T^{4} - 94076 T^{5} + 2318678 T^{6} - 5620212 T^{7} + 196672526 T^{8} - 5620212 p T^{9} + 2318678 p^{2} T^{10} - 94076 p^{3} T^{11} + 29608 p^{4} T^{12} - 1812 p^{5} T^{13} + 282 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 8 T + 616 T^{2} + 4244 T^{3} + 169700 T^{4} + 995260 T^{5} + 27290104 T^{6} + 133633280 T^{7} + 2801581334 T^{8} + 133633280 p T^{9} + 27290104 p^{2} T^{10} + 995260 p^{3} T^{11} + 169700 p^{4} T^{12} + 4244 p^{5} T^{13} + 616 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 12 T + 316 T^{2} + 2504 T^{3} + 36260 T^{4} + 40992 T^{5} + 689796 T^{6} - 28701716 T^{7} - 131086986 T^{8} - 28701716 p T^{9} + 689796 p^{2} T^{10} + 40992 p^{3} T^{11} + 36260 p^{4} T^{12} + 2504 p^{5} T^{13} + 316 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 16 T + 596 T^{2} - 6800 T^{3} + 152260 T^{4} - 1383968 T^{5} + 24247644 T^{6} - 186238592 T^{7} + 2748802518 T^{8} - 186238592 p T^{9} + 24247644 p^{2} T^{10} - 1383968 p^{3} T^{11} + 152260 p^{4} T^{12} - 6800 p^{5} T^{13} + 596 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.79752454289482616995357452928, −3.40511996940753958010081109851, −3.26657626457563681950991518597, −3.19972358635168019337874589509, −3.15841805328741432209865876059, −3.12477796909336906577067401104, −2.99835264138898912000984326483, −2.88268149853655992603567265294, −2.81360199195475935709795310286, −2.44963580199349348031386369385, −2.15098113762045182370763898782, −2.01939445412549633272278083207, −1.98682124437432689105064265350, −1.96996952547322643964347186666, −1.96223556634589965983488624934, −1.90225615557712198814826565915, −1.78599504935920333233772536024, −1.18551957697084490724709988211, −1.16256757955017448808660945585, −1.09839432668947993786859312568, −1.01806165169176687269283320691, −0.813349962401600924061871787594, −0.812976330931551710405763141763, −0.55676331421238503860253861745, −0.46977906764429856927438918322, 0.46977906764429856927438918322, 0.55676331421238503860253861745, 0.812976330931551710405763141763, 0.813349962401600924061871787594, 1.01806165169176687269283320691, 1.09839432668947993786859312568, 1.16256757955017448808660945585, 1.18551957697084490724709988211, 1.78599504935920333233772536024, 1.90225615557712198814826565915, 1.96223556634589965983488624934, 1.96996952547322643964347186666, 1.98682124437432689105064265350, 2.01939445412549633272278083207, 2.15098113762045182370763898782, 2.44963580199349348031386369385, 2.81360199195475935709795310286, 2.88268149853655992603567265294, 2.99835264138898912000984326483, 3.12477796909336906577067401104, 3.15841805328741432209865876059, 3.19972358635168019337874589509, 3.26657626457563681950991518597, 3.40511996940753958010081109851, 3.79752454289482616995357452928
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.