Dirichlet series
| L(s) = 1 | − 2·3-s + 4·5-s + 7-s + 3·9-s − 4·11-s − 4·13-s − 8·15-s − 3·17-s − 13·19-s − 2·21-s + 10·23-s + 17·25-s + 8·27-s + 8·29-s − 31-s + 8·33-s + 4·35-s + 8·37-s + 8·39-s + 18·41-s + 28·43-s + 12·45-s − 11·47-s + 6·49-s + 6·51-s + 53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.377·7-s + 9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s − 0.727·17-s − 2.98·19-s − 0.436·21-s + 2.08·23-s + 17/5·25-s + 1.53·27-s + 1.48·29-s − 0.179·31-s + 1.39·33-s + 0.676·35-s + 1.31·37-s + 1.28·39-s + 2.81·41-s + 4.26·43-s + 1.78·45-s − 1.60·47-s + 6/7·49-s + 0.840·51-s + 0.137·53-s − 2.15·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.77207\times 10^{7}\) |
| Root analytic conductor: | \(3.13649\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.206200096\) |
| \(L(\frac12)\) | \(\approx\) | \(8.206200096\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T - 5 T^{2} + T^{3} + 5 T^{4} + p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | |
| 11 | \( ( 1 + T + T^{2} )^{4} \) | |
| good | 3 | \( 1 + 2 T + T^{2} - 4 p T^{3} - 10 p T^{4} - 10 p T^{5} + 2 p^{2} T^{6} + 5 p^{3} T^{7} + 28 p^{2} T^{8} + 5 p^{4} T^{9} + 2 p^{4} T^{10} - 10 p^{4} T^{11} - 10 p^{5} T^{12} - 4 p^{6} T^{13} + p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 4 T - T^{2} + 22 T^{3} - 17 T^{4} + 18 T^{5} - 167 T^{6} - 44 p T^{7} + 1901 T^{8} - 44 p^{2} T^{9} - 167 p^{2} T^{10} + 18 p^{3} T^{11} - 17 p^{4} T^{12} + 22 p^{5} T^{13} - p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 + 2 T + 43 T^{2} + 69 T^{3} + 801 T^{4} + 69 p T^{5} + 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 3 T - 11 T^{2} + 48 T^{3} + 370 T^{4} + 150 T^{5} + 650 p T^{6} + 28071 T^{7} - 111155 T^{8} + 28071 p T^{9} + 650 p^{3} T^{10} + 150 p^{3} T^{11} + 370 p^{4} T^{12} + 48 p^{5} T^{13} - 11 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 13 T + 69 T^{2} + 72 T^{3} - 1044 T^{4} - 6546 T^{5} - 8662 T^{6} + 105335 T^{7} + 790941 T^{8} + 105335 p T^{9} - 8662 p^{2} T^{10} - 6546 p^{3} T^{11} - 1044 p^{4} T^{12} + 72 p^{5} T^{13} + 69 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 10 T - 13 T^{2} + 216 T^{3} + 85 p T^{4} - 9848 T^{5} - 43799 T^{6} + 30282 T^{7} + 1660447 T^{8} + 30282 p T^{9} - 43799 p^{2} T^{10} - 9848 p^{3} T^{11} + 85 p^{5} T^{12} + 216 p^{5} T^{13} - 13 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 - 4 T + 65 T^{2} - 361 T^{3} + 2299 T^{4} - 361 p T^{5} + 65 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 + T - 57 T^{2} - 154 T^{3} + 1424 T^{4} + 6 p^{2} T^{5} + 13054 T^{6} - 101243 T^{7} - 1081503 T^{8} - 101243 p T^{9} + 13054 p^{2} T^{10} + 6 p^{5} T^{11} + 1424 p^{4} T^{12} - 154 p^{5} T^{13} - 57 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T - 72 T^{2} + 480 T^{3} + 4770 T^{4} - 16368 T^{5} - 258784 T^{6} + 151544 T^{7} + 12108243 T^{8} + 151544 p T^{9} - 258784 p^{2} T^{10} - 16368 p^{3} T^{11} + 4770 p^{4} T^{12} + 480 p^{5} T^{13} - 72 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 9 T + 110 T^{2} - 850 T^{3} + 150 p T^{4} - 850 p T^{5} + 110 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 14 T + 235 T^{2} - 1905 T^{3} + 16560 T^{4} - 1905 p T^{5} + 235 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 11 T + 65 T^{2} + 1032 T^{3} + 7928 T^{4} + 33664 T^{5} + 11456 p T^{6} + 4118541 T^{7} + 17554669 T^{8} + 4118541 p T^{9} + 11456 p^{3} T^{10} + 33664 p^{3} T^{11} + 7928 p^{4} T^{12} + 1032 p^{5} T^{13} + 65 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - T - 151 T^{2} - 84 T^{3} + 12326 T^{4} + 14518 T^{5} - 779798 T^{6} - 393333 T^{7} + 42880405 T^{8} - 393333 p T^{9} - 779798 p^{2} T^{10} + 14518 p^{3} T^{11} + 12326 p^{4} T^{12} - 84 p^{5} T^{13} - 151 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 33 T + 496 T^{2} + 5431 T^{3} + 55765 T^{4} + 476926 T^{5} + 3209933 T^{6} + 22819873 T^{7} + 182469482 T^{8} + 22819873 p T^{9} + 3209933 p^{2} T^{10} + 476926 p^{3} T^{11} + 55765 p^{4} T^{12} + 5431 p^{5} T^{13} + 496 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 9 T - 19 T^{2} - 1502 T^{3} + 13831 T^{4} + 28265 T^{5} + 1112250 T^{6} - 9401651 T^{7} - 18552696 T^{8} - 9401651 p T^{9} + 1112250 p^{2} T^{10} + 28265 p^{3} T^{11} + 13831 p^{4} T^{12} - 1502 p^{5} T^{13} - 19 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 25 T + 225 T^{2} - 1382 T^{3} + 191 p T^{4} - 38403 T^{5} - 850094 T^{6} + 8442623 T^{7} - 47946114 T^{8} + 8442623 p T^{9} - 850094 p^{2} T^{10} - 38403 p^{3} T^{11} + 191 p^{5} T^{12} - 1382 p^{5} T^{13} + 225 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 3 T + 236 T^{2} + 492 T^{3} + 23340 T^{4} + 492 p T^{5} + 236 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 79 T^{2} - 86 T^{3} + 973 T^{4} + 6536 T^{5} + 427659 T^{6} - 269438 T^{7} - 34039683 T^{8} - 269438 p T^{9} + 427659 p^{2} T^{10} + 6536 p^{3} T^{11} + 973 p^{4} T^{12} - 86 p^{5} T^{13} - 79 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 - 28 T + 279 T^{2} - 1350 T^{3} + 12150 T^{4} - 161040 T^{5} + 827120 T^{6} + 107293 T^{7} - 15907704 T^{8} + 107293 p T^{9} + 827120 p^{2} T^{10} - 161040 p^{3} T^{11} + 12150 p^{4} T^{12} - 1350 p^{5} T^{13} + 279 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 5 T + 206 T^{2} + 982 T^{3} + 23660 T^{4} + 982 p T^{5} + 206 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 3 T - 209 T^{2} - 150 T^{3} + 20134 T^{4} - 33690 T^{5} - 2200286 T^{6} + 2222001 T^{7} + 244728697 T^{8} + 2222001 p T^{9} - 2200286 p^{2} T^{10} - 33690 p^{3} T^{11} + 20134 p^{4} T^{12} - 150 p^{5} T^{13} - 209 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 20 T + 403 T^{2} - 4961 T^{3} + 61051 T^{4} - 4961 p T^{5} + 403 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−4.26366044517993092306526240656, −4.22595356462340497847059290870, −4.12751348349294342360213648053, −3.72958902471867478528053967698, −3.59955459734078911678920968015, −3.48546472257027595229508566709, −3.22308524416347641758712578886, −3.11870686691586069998843844887, −3.10342217303506410126362370528, −2.78990708129728574701902862356, −2.75807296393225168371720616429, −2.55204277513786211415241651793, −2.41831607972094015790649915936, −2.32031843131000840079202209355, −2.29819004951425381747512745795, −2.16632103221284651471234788246, −2.13912481872094602978614318627, −1.70720351239590826507889529511, −1.44561089643196115470834536453, −1.36990931727632210773986748334, −1.02383531020239759713313240007, −0.913792783683382406522894114100, −0.790348570351158843037401224305, −0.61374812584920777486864902610, −0.32687896726665322131947839161, 0.32687896726665322131947839161, 0.61374812584920777486864902610, 0.790348570351158843037401224305, 0.913792783683382406522894114100, 1.02383531020239759713313240007, 1.36990931727632210773986748334, 1.44561089643196115470834536453, 1.70720351239590826507889529511, 2.13912481872094602978614318627, 2.16632103221284651471234788246, 2.29819004951425381747512745795, 2.32031843131000840079202209355, 2.41831607972094015790649915936, 2.55204277513786211415241651793, 2.75807296393225168371720616429, 2.78990708129728574701902862356, 3.10342217303506410126362370528, 3.11870686691586069998843844887, 3.22308524416347641758712578886, 3.48546472257027595229508566709, 3.59955459734078911678920968015, 3.72958902471867478528053967698, 4.12751348349294342360213648053, 4.22595356462340497847059290870, 4.26366044517993092306526240656
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.