Dirichlet series
| L(s) = 1 | − 4·3-s − 2·5-s − 2·7-s + 2·9-s + 8·15-s − 14·17-s + 12·19-s + 8·21-s − 8·23-s + 9·25-s + 28·27-s + 2·29-s + 18·31-s + 4·35-s + 14·37-s − 22·41-s + 20·43-s − 4·45-s + 10·47-s + 49-s + 56·51-s − 8·53-s − 48·57-s − 16·59-s + 14·61-s − 4·63-s − 26·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s + 2.06·15-s − 3.39·17-s + 2.75·19-s + 1.74·21-s − 1.66·23-s + 9/5·25-s + 5.38·27-s + 0.371·29-s + 3.23·31-s + 0.676·35-s + 2.30·37-s − 3.43·41-s + 3.04·43-s − 0.596·45-s + 1.45·47-s + 1/7·49-s + 7.84·51-s − 1.09·53-s − 6.35·57-s − 2.08·59-s + 1.79·61-s − 0.503·63-s − 3.17·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 7^{8} \cdot 41^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.98599\times 10^{7}\) |
| Root analytic conductor: | \(3.02767\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 7^{8} \cdot 41^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1497851859\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1497851859\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 41 | \( 1 + 22 T + 253 T^{2} + 1904 T^{3} + 12465 T^{4} + 1904 p T^{5} + 253 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( ( 1 + 2 T + 5 T^{2} + p T^{4} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 + 2 T - p T^{2} - 18 T^{3} + 4 T^{4} + 88 T^{5} + 17 p T^{6} - 132 T^{7} - 249 T^{8} - 132 p T^{9} + 17 p^{3} T^{10} + 88 p^{3} T^{11} + 4 p^{4} T^{12} - 18 p^{5} T^{13} - p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 4 T^{2} - 60 T^{3} + 65 T^{4} + 280 T^{5} + 1444 T^{6} - 3860 T^{7} - 8131 T^{8} - 3860 p T^{9} + 1444 p^{2} T^{10} + 280 p^{3} T^{11} + 65 p^{4} T^{12} - 60 p^{5} T^{13} - 4 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T^{2} - 20 T^{3} + 175 T^{4} - 600 T^{5} + 452 T^{6} - 7920 T^{7} + 29669 T^{8} - 7920 p T^{9} + 452 p^{2} T^{10} - 600 p^{3} T^{11} + 175 p^{4} T^{12} - 20 p^{5} T^{13} + 2 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 14 T + 105 T^{2} + 584 T^{3} + 2699 T^{4} + 8732 T^{5} + 13771 T^{6} - 18642 T^{7} - 168304 T^{8} - 18642 p T^{9} + 13771 p^{2} T^{10} + 8732 p^{3} T^{11} + 2699 p^{4} T^{12} + 584 p^{5} T^{13} + 105 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 12 T + 60 T^{2} - 240 T^{3} + 1065 T^{4} - 5796 T^{5} + 37052 T^{6} - 178740 T^{7} + 730485 T^{8} - 178740 p T^{9} + 37052 p^{2} T^{10} - 5796 p^{3} T^{11} + 1065 p^{4} T^{12} - 240 p^{5} T^{13} + 60 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 8 T + 10 T^{2} + 52 T^{3} + 1099 T^{4} + 3464 T^{5} + 3184 T^{6} + 48464 T^{7} + 389301 T^{8} + 48464 p T^{9} + 3184 p^{2} T^{10} + 3464 p^{3} T^{11} + 1099 p^{4} T^{12} + 52 p^{5} T^{13} + 10 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 2 T - 63 T^{2} + 202 T^{3} + 1212 T^{4} - 7380 T^{5} + 33223 T^{6} + 108476 T^{7} - 2109345 T^{8} + 108476 p T^{9} + 33223 p^{2} T^{10} - 7380 p^{3} T^{11} + 1212 p^{4} T^{12} + 202 p^{5} T^{13} - 63 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 18 T + 163 T^{2} - 984 T^{3} + 147 p T^{4} - 9360 T^{5} - 92803 T^{6} + 1220022 T^{7} - 8202220 T^{8} + 1220022 p T^{9} - 92803 p^{2} T^{10} - 9360 p^{3} T^{11} + 147 p^{5} T^{12} - 984 p^{5} T^{13} + 163 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 14 T - 19 T^{2} + 1278 T^{3} - 4204 T^{4} - 58132 T^{5} + 453635 T^{6} + 928360 T^{7} - 21597513 T^{8} + 928360 p T^{9} + 453635 p^{2} T^{10} - 58132 p^{3} T^{11} - 4204 p^{4} T^{12} + 1278 p^{5} T^{13} - 19 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 20 T + 216 T^{2} - 2300 T^{3} + 25357 T^{4} - 220220 T^{5} + 1629828 T^{6} - 12477700 T^{7} + 89951005 T^{8} - 12477700 p T^{9} + 1629828 p^{2} T^{10} - 220220 p^{3} T^{11} + 25357 p^{4} T^{12} - 2300 p^{5} T^{13} + 216 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 73 T^{2} - 700 T^{3} + 7185 T^{4} - 52700 T^{5} + 351543 T^{6} - 2556170 T^{7} + 20003404 T^{8} - 2556170 p T^{9} + 351543 p^{2} T^{10} - 52700 p^{3} T^{11} + 7185 p^{4} T^{12} - 700 p^{5} T^{13} + 73 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 8 T + 14 T^{2} - 324 T^{3} - 5069 T^{4} - 41016 T^{5} + 440 T^{6} + 1680080 T^{7} + 12984517 T^{8} + 1680080 p T^{9} + 440 p^{2} T^{10} - 41016 p^{3} T^{11} - 5069 p^{4} T^{12} - 324 p^{5} T^{13} + 14 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 16 T - 4 T^{2} - 940 T^{3} + 353 T^{4} + 77960 T^{5} + 675924 T^{6} - 605924 T^{7} - 43234227 T^{8} - 605924 p T^{9} + 675924 p^{2} T^{10} + 77960 p^{3} T^{11} + 353 p^{4} T^{12} - 940 p^{5} T^{13} - 4 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 14 T + 143 T^{2} - 1314 T^{3} + 14732 T^{4} - 96040 T^{5} + 562217 T^{6} - 3545192 T^{7} + 38081055 T^{8} - 3545192 p T^{9} + 562217 p^{2} T^{10} - 96040 p^{3} T^{11} + 14732 p^{4} T^{12} - 1314 p^{5} T^{13} + 143 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 26 T + 165 T^{2} - 934 T^{3} - 11966 T^{4} - 2692 T^{5} + 380841 T^{6} + 6011632 T^{7} + 71745151 T^{8} + 6011632 p T^{9} + 380841 p^{2} T^{10} - 2692 p^{3} T^{11} - 11966 p^{4} T^{12} - 934 p^{5} T^{13} + 165 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 14 T + 53 T^{2} - 774 T^{3} + 13802 T^{4} - 93020 T^{5} + 740217 T^{6} - 8831712 T^{7} + 79046935 T^{8} - 8831712 p T^{9} + 740217 p^{2} T^{10} - 93020 p^{3} T^{11} + 13802 p^{4} T^{12} - 774 p^{5} T^{13} + 53 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( ( 1 + 20 T + 328 T^{2} + 3500 T^{3} + 34814 T^{4} + 3500 p T^{5} + 328 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 36 T + 636 T^{2} + 7380 T^{3} + 69605 T^{4} + 7380 p T^{5} + 636 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 83 | \( ( 1 + 4 T + 172 T^{2} + 468 T^{3} + 16133 T^{4} + 468 p T^{5} + 172 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 2 T - 73 T^{2} + 1482 T^{3} + 9572 T^{4} + 55600 T^{5} - 505887 T^{6} + 21624 p T^{7} + 2722175 p T^{8} + 21624 p^{2} T^{9} - 505887 p^{2} T^{10} + 55600 p^{3} T^{11} + 9572 p^{4} T^{12} + 1482 p^{5} T^{13} - 73 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 18 T - 39 T^{2} + 3760 T^{3} - 30189 T^{4} - 124364 T^{5} + 2955355 T^{6} - 6960594 T^{7} - 102088448 T^{8} - 6960594 p T^{9} + 2955355 p^{2} T^{10} - 124364 p^{3} T^{11} - 30189 p^{4} T^{12} + 3760 p^{5} T^{13} - 39 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.32690825351213170940468345986, −4.29909684626075289159972046584, −4.27836161424539883767364793373, −3.87060505925062891708548916880, −3.51824029316075219685802827578, −3.49602915746879102966148250059, −3.30411039497061310453482974413, −3.25823192314921846460125891322, −3.18406185918895366993738209715, −2.79638511309419516364681011421, −2.77611811465413412057021161114, −2.67925333545914113920040347442, −2.66832498128976623753852295138, −2.65620763015777520873515425166, −2.48744168446410619072041973059, −2.14267926132513298600084474488, −1.87108249982408437847126942528, −1.54194563042002239071718295632, −1.29176710308876266024530722089, −1.27542362747379736586551225708, −1.20262544711054714880500540580, −0.981991061905075289455924592806, −0.54592045003988424161862029109, −0.22472562309755801783440905706, −0.19072161721090562544673347328, 0.19072161721090562544673347328, 0.22472562309755801783440905706, 0.54592045003988424161862029109, 0.981991061905075289455924592806, 1.20262544711054714880500540580, 1.27542362747379736586551225708, 1.29176710308876266024530722089, 1.54194563042002239071718295632, 1.87108249982408437847126942528, 2.14267926132513298600084474488, 2.48744168446410619072041973059, 2.65620763015777520873515425166, 2.66832498128976623753852295138, 2.67925333545914113920040347442, 2.77611811465413412057021161114, 2.79638511309419516364681011421, 3.18406185918895366993738209715, 3.25823192314921846460125891322, 3.30411039497061310453482974413, 3.49602915746879102966148250059, 3.51824029316075219685802827578, 3.87060505925062891708548916880, 4.27836161424539883767364793373, 4.29909684626075289159972046584, 4.32690825351213170940468345986
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.