Dirichlet series
| L(s) = 1 | + 5-s − 9·9-s + 9·11-s − 4·17-s − 8·19-s + 25·23-s − 12·25-s − 27-s + 6·29-s + 13·37-s + 16·41-s + 17·43-s − 9·45-s + 24·47-s + 2·53-s + 9·55-s − 2·59-s + 13·61-s + 2·67-s + 10·71-s − 5·73-s + 16·79-s + 33·81-s + 43·83-s − 4·85-s − 8·89-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 3·9-s + 2.71·11-s − 0.970·17-s − 1.83·19-s + 5.21·23-s − 2.39·25-s − 0.192·27-s + 1.11·29-s + 2.13·37-s + 2.49·41-s + 2.59·43-s − 1.34·45-s + 3.50·47-s + 0.274·53-s + 1.21·55-s − 0.260·59-s + 1.66·61-s + 0.244·67-s + 1.18·71-s − 0.585·73-s + 1.80·79-s + 11/3·81-s + 4.71·83-s − 0.433·85-s − 0.847·89-s − 0.820·95-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 7^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56507\times 10^{14}\) |
| Root analytic conductor: | \(7.71184\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 7^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(54.21697414\) |
| \(L(\frac12)\) | \(\approx\) | \(54.21697414\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 19 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + p^{2} T^{2} + T^{3} + 16 p T^{4} + 11 T^{5} + 203 T^{6} + 46 T^{7} + 694 T^{8} + 46 p T^{9} + 203 p^{2} T^{10} + 11 p^{3} T^{11} + 16 p^{5} T^{12} + p^{5} T^{13} + p^{8} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - T + 13 T^{2} - 26 T^{3} + 112 T^{4} - 248 T^{5} + 851 T^{6} - 1537 T^{7} + 5006 T^{8} - 1537 p T^{9} + 851 p^{2} T^{10} - 248 p^{3} T^{11} + 112 p^{4} T^{12} - 26 p^{5} T^{13} + 13 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 9 T + 82 T^{2} - 459 T^{3} + 2417 T^{4} - 9780 T^{5} + 38622 T^{6} - 128886 T^{7} + 453764 T^{8} - 128886 p T^{9} + 38622 p^{2} T^{10} - 9780 p^{3} T^{11} + 2417 p^{4} T^{12} - 459 p^{5} T^{13} + 82 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 63 T^{2} - T^{3} + 1844 T^{4} - 265 T^{5} + 34701 T^{6} - 9002 T^{7} + 498638 T^{8} - 9002 p T^{9} + 34701 p^{2} T^{10} - 265 p^{3} T^{11} + 1844 p^{4} T^{12} - p^{5} T^{13} + 63 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T + 58 T^{2} + 185 T^{3} + 121 p T^{4} + 6234 T^{5} + 53760 T^{6} + 139113 T^{7} + 1017128 T^{8} + 139113 p T^{9} + 53760 p^{2} T^{10} + 6234 p^{3} T^{11} + 121 p^{5} T^{12} + 185 p^{5} T^{13} + 58 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 25 T + 413 T^{2} - 4867 T^{3} + 46668 T^{4} - 369316 T^{5} + 2515264 T^{6} - 14747354 T^{7} + 75826491 T^{8} - 14747354 p T^{9} + 2515264 p^{2} T^{10} - 369316 p^{3} T^{11} + 46668 p^{4} T^{12} - 4867 p^{5} T^{13} + 413 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 6 T + 127 T^{2} - 889 T^{3} + 8908 T^{4} - 58681 T^{5} + 443085 T^{6} - 2388316 T^{7} + 15502590 T^{8} - 2388316 p T^{9} + 443085 p^{2} T^{10} - 58681 p^{3} T^{11} + 8908 p^{4} T^{12} - 889 p^{5} T^{13} + 127 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 148 T^{2} + 116 T^{3} + 10252 T^{4} + 17596 T^{5} + 457996 T^{6} + 1076104 T^{7} + 15671846 T^{8} + 1076104 p T^{9} + 457996 p^{2} T^{10} + 17596 p^{3} T^{11} + 10252 p^{4} T^{12} + 116 p^{5} T^{13} + 148 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 13 T + 207 T^{2} - 1750 T^{3} + 16402 T^{4} - 112836 T^{5} + 816117 T^{6} - 5071053 T^{7} + 32170002 T^{8} - 5071053 p T^{9} + 816117 p^{2} T^{10} - 112836 p^{3} T^{11} + 16402 p^{4} T^{12} - 1750 p^{5} T^{13} + 207 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 2 T + p T^{2} )^{8} \) | |
| 43 | \( 1 - 17 T + 355 T^{2} - 4036 T^{3} + 49646 T^{4} - 429858 T^{5} + 3936957 T^{6} - 642057 p T^{7} + 204548050 T^{8} - 642057 p^{2} T^{9} + 3936957 p^{2} T^{10} - 429858 p^{3} T^{11} + 49646 p^{4} T^{12} - 4036 p^{5} T^{13} + 355 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 24 T + 392 T^{2} - 4364 T^{3} + 37602 T^{4} - 233364 T^{5} + 1036208 T^{6} - 2240436 T^{7} + 2928963 T^{8} - 2240436 p T^{9} + 1036208 p^{2} T^{10} - 233364 p^{3} T^{11} + 37602 p^{4} T^{12} - 4364 p^{5} T^{13} + 392 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T + 129 T^{2} + 37 T^{3} + 10574 T^{4} - 3559 T^{5} + 772851 T^{6} - 539736 T^{7} + 43318986 T^{8} - 539736 p T^{9} + 772851 p^{2} T^{10} - 3559 p^{3} T^{11} + 10574 p^{4} T^{12} + 37 p^{5} T^{13} + 129 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 2 T + 161 T^{2} + 671 T^{3} + 14184 T^{4} + 82025 T^{5} + 987807 T^{6} + 6008672 T^{7} + 60662110 T^{8} + 6008672 p T^{9} + 987807 p^{2} T^{10} + 82025 p^{3} T^{11} + 14184 p^{4} T^{12} + 671 p^{5} T^{13} + 161 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 13 T + 330 T^{2} - 3221 T^{3} + 51309 T^{4} - 421256 T^{5} + 5202698 T^{6} - 36815298 T^{7} + 375128276 T^{8} - 36815298 p T^{9} + 5202698 p^{2} T^{10} - 421256 p^{3} T^{11} + 51309 p^{4} T^{12} - 3221 p^{5} T^{13} + 330 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 2 T + 453 T^{2} - 869 T^{3} + 94066 T^{4} - 164739 T^{5} + 11710995 T^{6} - 17822484 T^{7} + 956952666 T^{8} - 17822484 p T^{9} + 11710995 p^{2} T^{10} - 164739 p^{3} T^{11} + 94066 p^{4} T^{12} - 869 p^{5} T^{13} + 453 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 10 T + 328 T^{2} - 2494 T^{3} + 51564 T^{4} - 327342 T^{5} + 78728 p T^{6} - 31264890 T^{7} + 457121126 T^{8} - 31264890 p T^{9} + 78728 p^{3} T^{10} - 327342 p^{3} T^{11} + 51564 p^{4} T^{12} - 2494 p^{5} T^{13} + 328 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 5 T + 227 T^{2} + 759 T^{3} + 19426 T^{4} - 26030 T^{5} + 597516 T^{6} - 13022320 T^{7} - 270615 T^{8} - 13022320 p T^{9} + 597516 p^{2} T^{10} - 26030 p^{3} T^{11} + 19426 p^{4} T^{12} + 759 p^{5} T^{13} + 227 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 412 T^{2} - 5444 T^{3} + 81524 T^{4} - 930572 T^{5} + 10484580 T^{6} - 103560600 T^{7} + 965277206 T^{8} - 103560600 p T^{9} + 10484580 p^{2} T^{10} - 930572 p^{3} T^{11} + 81524 p^{4} T^{12} - 5444 p^{5} T^{13} + 412 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 43 T + 1290 T^{2} - 27837 T^{3} + 495581 T^{4} - 7318764 T^{5} + 93657586 T^{6} - 1035827238 T^{7} + 10099433844 T^{8} - 1035827238 p T^{9} + 93657586 p^{2} T^{10} - 7318764 p^{3} T^{11} + 495581 p^{4} T^{12} - 27837 p^{5} T^{13} + 1290 p^{6} T^{14} - 43 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 8 T + 448 T^{2} + 3580 T^{3} + 98092 T^{4} + 710908 T^{5} + 13956896 T^{6} + 88249184 T^{7} + 16134934 p T^{8} + 88249184 p T^{9} + 13956896 p^{2} T^{10} + 710908 p^{3} T^{11} + 98092 p^{4} T^{12} + 3580 p^{5} T^{13} + 448 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 12 T + 608 T^{2} - 5812 T^{3} + 170620 T^{4} - 1355452 T^{5} + 29452384 T^{6} - 196855172 T^{7} + 3436368006 T^{8} - 196855172 p T^{9} + 29452384 p^{2} T^{10} - 1355452 p^{3} T^{11} + 170620 p^{4} T^{12} - 5812 p^{5} T^{13} + 608 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.22260856585470996167753611631, −2.86406381464801700957671847416, −2.83501064254502908029822297511, −2.80355696791682877384286866283, −2.71853102212742251377170461180, −2.69888899248272404534214859181, −2.56400462536362974240929140584, −2.55444892489361382595997187067, −2.50036986497584848428373178316, −2.10753831087071319223293367197, −2.07631028673759309496533996353, −1.95071069959175833057308310539, −1.93229292745267334452164301881, −1.91699894144696212685307326716, −1.72671915796987837296735388566, −1.46878919362282994555212454353, −1.23360391927338005960724201367, −1.15856185298537826003382584659, −0.882216032996802823625228410308, −0.794978453017742340859452561448, −0.75223176940564700588634755394, −0.70334340376276612674475086896, −0.60191392593272752868399812468, −0.53411161804527664603932231875, −0.28350124694912902262576559109, 0.28350124694912902262576559109, 0.53411161804527664603932231875, 0.60191392593272752868399812468, 0.70334340376276612674475086896, 0.75223176940564700588634755394, 0.794978453017742340859452561448, 0.882216032996802823625228410308, 1.15856185298537826003382584659, 1.23360391927338005960724201367, 1.46878919362282994555212454353, 1.72671915796987837296735388566, 1.91699894144696212685307326716, 1.93229292745267334452164301881, 1.95071069959175833057308310539, 2.07631028673759309496533996353, 2.10753831087071319223293367197, 2.50036986497584848428373178316, 2.55444892489361382595997187067, 2.56400462536362974240929140584, 2.69888899248272404534214859181, 2.71853102212742251377170461180, 2.80355696791682877384286866283, 2.83501064254502908029822297511, 2.86406381464801700957671847416, 3.22260856585470996167753611631
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.