Dirichlet series
| L(s) = 1 | + 2-s + 4·3-s − 4·4-s + 10·5-s + 4·6-s + 8·7-s − 5·8-s + 3·9-s + 10·10-s − 16·12-s − 6·13-s + 8·14-s + 40·15-s + 4·16-s − 5·17-s + 3·18-s − 13·19-s − 40·20-s + 32·21-s + 16·23-s − 20·24-s + 38·25-s − 6·26-s − 10·27-s − 32·28-s + 9·29-s + 40·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s − 2·4-s + 4.47·5-s + 1.63·6-s + 3.02·7-s − 1.76·8-s + 9-s + 3.16·10-s − 4.61·12-s − 1.66·13-s + 2.13·14-s + 10.3·15-s + 16-s − 1.21·17-s + 0.707·18-s − 2.98·19-s − 8.94·20-s + 6.98·21-s + 3.33·23-s − 4.08·24-s + 38/5·25-s − 1.17·26-s − 1.92·27-s − 6.04·28-s + 1.67·29-s + 7.30·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.37808\times 10^{6}\) |
| Root analytic conductor: | \(2.60064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(42.75211995\) |
| \(L(\frac12)\) | \(\approx\) | \(42.75211995\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T )^{8} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T + 5 T^{2} - p^{2} T^{3} + 15 T^{4} - 7 p T^{5} + 19 p T^{6} - 15 p T^{7} + 75 T^{8} - 15 p^{2} T^{9} + 19 p^{3} T^{10} - 7 p^{4} T^{11} + 15 p^{4} T^{12} - p^{7} T^{13} + 5 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 23 p T^{4} - 44 p T^{5} + 271 T^{6} - 166 p T^{7} + 940 T^{8} - 166 p^{2} T^{9} + 271 p^{2} T^{10} - 44 p^{4} T^{11} + 23 p^{5} T^{12} - 10 p^{6} T^{13} + 13 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 - 2 p T + 62 T^{2} - 283 T^{3} + 43 p^{2} T^{4} - 3513 T^{5} + 2046 p T^{6} - 26646 T^{7} + 62784 T^{8} - 26646 p T^{9} + 2046 p^{3} T^{10} - 3513 p^{3} T^{11} + 43 p^{6} T^{12} - 283 p^{5} T^{13} + 62 p^{6} T^{14} - 2 p^{8} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 6 T + 68 T^{2} + 285 T^{3} + 163 p T^{4} + 7443 T^{5} + 44016 T^{6} + 130942 T^{7} + 656120 T^{8} + 130942 p T^{9} + 44016 p^{2} T^{10} + 7443 p^{3} T^{11} + 163 p^{5} T^{12} + 285 p^{5} T^{13} + 68 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 5 T + 99 T^{2} + 384 T^{3} + 4383 T^{4} + 13616 T^{5} + 119813 T^{6} + 309659 T^{7} + 2344384 T^{8} + 309659 p T^{9} + 119813 p^{2} T^{10} + 13616 p^{3} T^{11} + 4383 p^{4} T^{12} + 384 p^{5} T^{13} + 99 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 13 T + 167 T^{2} + 1424 T^{3} + 11403 T^{4} + 72606 T^{5} + 434133 T^{6} + 2182431 T^{7} + 10302272 T^{8} + 2182431 p T^{9} + 434133 p^{2} T^{10} + 72606 p^{3} T^{11} + 11403 p^{4} T^{12} + 1424 p^{5} T^{13} + 167 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 40290 p T^{6} - 5230592 T^{7} + 26816927 T^{8} - 5230592 p T^{9} + 40290 p^{3} T^{10} - 141712 p^{3} T^{11} + 19567 p^{4} T^{12} - 2234 p^{5} T^{13} + 224 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 9 T + 187 T^{2} - 1382 T^{3} + 16028 T^{4} - 97473 T^{5} + 828470 T^{6} - 4208558 T^{7} + 28798593 T^{8} - 4208558 p T^{9} + 828470 p^{2} T^{10} - 97473 p^{3} T^{11} + 16028 p^{4} T^{12} - 1382 p^{5} T^{13} + 187 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 9 T + 142 T^{2} - 963 T^{3} + 9327 T^{4} - 55589 T^{5} + 14166 p T^{6} - 2365595 T^{7} + 15862880 T^{8} - 2365595 p T^{9} + 14166 p^{3} T^{10} - 55589 p^{3} T^{11} + 9327 p^{4} T^{12} - 963 p^{5} T^{13} + 142 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 7 T + 231 T^{2} - 1552 T^{3} + 24962 T^{4} - 154999 T^{5} + 1656240 T^{6} - 9060906 T^{7} + 73868707 T^{8} - 9060906 p T^{9} + 1656240 p^{2} T^{10} - 154999 p^{3} T^{11} + 24962 p^{4} T^{12} - 1552 p^{5} T^{13} + 231 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 10 T + 261 T^{2} + 2176 T^{3} + 30891 T^{4} + 221954 T^{5} + 54771 p T^{6} + 13854084 T^{7} + 110613464 T^{8} + 13854084 p T^{9} + 54771 p^{3} T^{10} + 221954 p^{3} T^{11} + 30891 p^{4} T^{12} + 2176 p^{5} T^{13} + 261 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 6271668 p T^{9} + 2007760 p^{2} T^{10} + 98508 p^{3} T^{11} + 27462 p^{4} T^{12} + 936 p^{5} T^{13} + 239 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 16 T + 320 T^{2} - 3319 T^{3} + 41705 T^{4} - 344729 T^{5} + 3361468 T^{6} - 23054530 T^{7} + 186100580 T^{8} - 23054530 p T^{9} + 3361468 p^{2} T^{10} - 344729 p^{3} T^{11} + 41705 p^{4} T^{12} - 3319 p^{5} T^{13} + 320 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 37 T + 857 T^{2} - 14496 T^{3} + 199396 T^{4} - 2305797 T^{5} + 23105040 T^{6} - 202528658 T^{7} + 1568879259 T^{8} - 202528658 p T^{9} + 23105040 p^{2} T^{10} - 2305797 p^{3} T^{11} + 199396 p^{4} T^{12} - 14496 p^{5} T^{13} + 857 p^{6} T^{14} - 37 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - T + 332 T^{2} - 463 T^{3} + 53293 T^{4} - 80577 T^{5} + 5426300 T^{6} - 7692267 T^{7} + 381400788 T^{8} - 7692267 p T^{9} + 5426300 p^{2} T^{10} - 80577 p^{3} T^{11} + 53293 p^{4} T^{12} - 463 p^{5} T^{13} + 332 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 19 T + 492 T^{2} - 7043 T^{3} + 104067 T^{4} - 1170549 T^{5} + 12549376 T^{6} - 113336285 T^{7} + 952230800 T^{8} - 113336285 p T^{9} + 12549376 p^{2} T^{10} - 1170549 p^{3} T^{11} + 104067 p^{4} T^{12} - 7043 p^{5} T^{13} + 492 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 150444400 p T^{9} + 15702408 p^{2} T^{10} + 1415171 p^{3} T^{11} + 119005 p^{4} T^{12} + 7751 p^{5} T^{13} + 520 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 13 T + 398 T^{2} - 3457 T^{3} + 57783 T^{4} - 323737 T^{5} + 4210848 T^{6} - 14543752 T^{7} + 249426533 T^{8} - 14543752 p T^{9} + 4210848 p^{2} T^{10} - 323737 p^{3} T^{11} + 57783 p^{4} T^{12} - 3457 p^{5} T^{13} + 398 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 25 T + 616 T^{2} + 9963 T^{3} + 152093 T^{4} + 1861173 T^{5} + 21394652 T^{6} + 209466223 T^{7} + 1925408604 T^{8} + 209466223 p T^{9} + 21394652 p^{2} T^{10} + 1861173 p^{3} T^{11} + 152093 p^{4} T^{12} + 9963 p^{5} T^{13} + 616 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 337 T^{2} + 1005 T^{3} + 57248 T^{4} + 275700 T^{5} + 6743044 T^{6} + 36745800 T^{7} + 601257265 T^{8} + 36745800 p T^{9} + 6743044 p^{2} T^{10} + 275700 p^{3} T^{11} + 57248 p^{4} T^{12} + 1005 p^{5} T^{13} + 337 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 + 25 T + 786 T^{2} + 13983 T^{3} + 250863 T^{4} + 3398873 T^{5} + 43717402 T^{6} + 464690723 T^{7} + 4604658344 T^{8} + 464690723 p T^{9} + 43717402 p^{2} T^{10} + 3398873 p^{3} T^{11} + 250863 p^{4} T^{12} + 13983 p^{5} T^{13} + 786 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 737152 p T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 731972999 p T^{9} + 737152 p^{3} T^{10} - 5152639 p^{3} T^{11} + 357823 p^{4} T^{12} - 20901 p^{5} T^{13} + 1032 p^{6} T^{14} - 37 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 15 T + 554 T^{2} - 7041 T^{3} + 141913 T^{4} - 1604409 T^{5} + 23035698 T^{6} - 230491951 T^{7} + 2630434004 T^{8} - 230491951 p T^{9} + 23035698 p^{2} T^{10} - 1604409 p^{3} T^{11} + 141913 p^{4} T^{12} - 7041 p^{5} T^{13} + 554 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.45908197828923935890090244784, −4.35516531437021935461518048227, −4.26685292618576175685198773340, −3.98727601889932024494652120202, −3.98710731060637845408335037659, −3.84779786636570692971909051084, −3.66769066806464202695774866371, −3.40008699568525865567821919817, −3.14544369580271069295246989923, −2.93850061063838819540398769505, −2.92198446594862392218162626649, −2.67312231773154155041049442055, −2.62659087405395493953217753664, −2.60404780516992387191230468659, −2.33360035185413230789288829731, −2.16901827663704131159076979811, −2.10653354956985356270880218676, −2.02658795804911188922133546710, −2.01583185687407529096482451557, −1.73173013630523210450813240316, −1.52521749126764786855776704973, −1.08269303412096328106029741899, −1.01459601045708145917534886069, −0.76670312481976986468045789396, −0.41389004845033337378477967177, 0.41389004845033337378477967177, 0.76670312481976986468045789396, 1.01459601045708145917534886069, 1.08269303412096328106029741899, 1.52521749126764786855776704973, 1.73173013630523210450813240316, 2.01583185687407529096482451557, 2.02658795804911188922133546710, 2.10653354956985356270880218676, 2.16901827663704131159076979811, 2.33360035185413230789288829731, 2.60404780516992387191230468659, 2.62659087405395493953217753664, 2.67312231773154155041049442055, 2.92198446594862392218162626649, 2.93850061063838819540398769505, 3.14544369580271069295246989923, 3.40008699568525865567821919817, 3.66769066806464202695774866371, 3.84779786636570692971909051084, 3.98710731060637845408335037659, 3.98727601889932024494652120202, 4.26685292618576175685198773340, 4.35516531437021935461518048227, 4.45908197828923935890090244784
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.