Dirichlet series
| L(s) = 1 | − 2·2-s − 8·3-s − 4·4-s + 2·5-s + 16·6-s + 10·8-s + 36·9-s − 4·10-s − 2·11-s + 32·12-s + 4·13-s − 16·15-s + 6·16-s + 8·17-s − 72·18-s + 6·19-s − 8·20-s + 4·22-s − 12·23-s − 80·24-s − 20·25-s − 8·26-s − 120·27-s − 4·29-s + 32·30-s − 10·31-s − 20·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.61·3-s − 2·4-s + 0.894·5-s + 6.53·6-s + 3.53·8-s + 12·9-s − 1.26·10-s − 0.603·11-s + 9.23·12-s + 1.10·13-s − 4.13·15-s + 3/2·16-s + 1.94·17-s − 16.9·18-s + 1.37·19-s − 1.78·20-s + 0.852·22-s − 2.50·23-s − 16.3·24-s − 4·25-s − 1.56·26-s − 23.0·27-s − 0.742·29-s + 5.84·30-s − 1.79·31-s − 3.53·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 7^{16} \cdot 41^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.87757\times 10^{13}\) |
| Root analytic conductor: | \(6.93727\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{8} \cdot 7^{16} \cdot 41^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{8} \) |
| 7 | \( 1 \) | |
| 41 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( 1 + p T + p^{3} T^{2} + 7 p T^{3} + 17 p T^{4} + 13 p^{2} T^{5} + 51 p T^{6} + 67 p T^{7} + 231 T^{8} + 67 p^{2} T^{9} + 51 p^{3} T^{10} + 13 p^{5} T^{11} + 17 p^{5} T^{12} + 7 p^{6} T^{13} + p^{9} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 2 T + 24 T^{2} - 44 T^{3} + 306 T^{4} - 502 T^{5} + 2552 T^{6} - 3658 T^{7} + 15067 T^{8} - 3658 p T^{9} + 2552 p^{2} T^{10} - 502 p^{3} T^{11} + 306 p^{4} T^{12} - 44 p^{5} T^{13} + 24 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T + 48 T^{2} + 34 T^{3} + 1092 T^{4} + 6 p T^{5} + 18272 T^{6} - 654 T^{7} + 235526 T^{8} - 654 p T^{9} + 18272 p^{2} T^{10} + 6 p^{4} T^{11} + 1092 p^{4} T^{12} + 34 p^{5} T^{13} + 48 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 74 T^{2} - 272 T^{3} + 2722 T^{4} - 8772 T^{5} + 62812 T^{6} - 173226 T^{7} + 981631 T^{8} - 173226 p T^{9} + 62812 p^{2} T^{10} - 8772 p^{3} T^{11} + 2722 p^{4} T^{12} - 272 p^{5} T^{13} + 74 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 8 T + 82 T^{2} - 492 T^{3} + 3436 T^{4} - 16884 T^{5} + 90854 T^{6} - 386276 T^{7} + 1789282 T^{8} - 386276 p T^{9} + 90854 p^{2} T^{10} - 16884 p^{3} T^{11} + 3436 p^{4} T^{12} - 492 p^{5} T^{13} + 82 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 6 T + 128 T^{2} - 600 T^{3} + 7132 T^{4} - 27040 T^{5} + 236448 T^{6} - 746214 T^{7} + 5330986 T^{8} - 746214 p T^{9} + 236448 p^{2} T^{10} - 27040 p^{3} T^{11} + 7132 p^{4} T^{12} - 600 p^{5} T^{13} + 128 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 12 T + 144 T^{2} + 1084 T^{3} + 7818 T^{4} + 43406 T^{5} + 238114 T^{6} + 1120130 T^{7} + 5638293 T^{8} + 1120130 p T^{9} + 238114 p^{2} T^{10} + 43406 p^{3} T^{11} + 7818 p^{4} T^{12} + 1084 p^{5} T^{13} + 144 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 4 T + 88 T^{2} + 260 T^{3} + 3598 T^{4} + 14282 T^{5} + 124596 T^{6} + 23450 p T^{7} + 135775 p T^{8} + 23450 p^{2} T^{9} + 124596 p^{2} T^{10} + 14282 p^{3} T^{11} + 3598 p^{4} T^{12} + 260 p^{5} T^{13} + 88 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 10 T + 242 T^{2} + 1916 T^{3} + 25404 T^{4} + 163008 T^{5} + 1534410 T^{6} + 8038430 T^{7} + 58878402 T^{8} + 8038430 p T^{9} + 1534410 p^{2} T^{10} + 163008 p^{3} T^{11} + 25404 p^{4} T^{12} + 1916 p^{5} T^{13} + 242 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 20 T + 284 T^{2} + 2844 T^{3} + 24582 T^{4} + 175636 T^{5} + 1169868 T^{6} + 7092876 T^{7} + 43912531 T^{8} + 7092876 p T^{9} + 1169868 p^{2} T^{10} + 175636 p^{3} T^{11} + 24582 p^{4} T^{12} + 2844 p^{5} T^{13} + 284 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T + 260 T^{2} + 1912 T^{3} + 30760 T^{4} + 209304 T^{5} + 2238648 T^{6} + 13769892 T^{7} + 113280482 T^{8} + 13769892 p T^{9} + 2238648 p^{2} T^{10} + 209304 p^{3} T^{11} + 30760 p^{4} T^{12} + 1912 p^{5} T^{13} + 260 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 24 T + 368 T^{2} - 3882 T^{3} + 36588 T^{4} - 298974 T^{5} + 2337676 T^{6} - 16192122 T^{7} + 113653485 T^{8} - 16192122 p T^{9} + 2337676 p^{2} T^{10} - 298974 p^{3} T^{11} + 36588 p^{4} T^{12} - 3882 p^{5} T^{13} + 368 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 36 T + 812 T^{2} + 12938 T^{3} + 167990 T^{4} + 1830390 T^{5} + 17627940 T^{6} + 150285932 T^{7} + 1157498979 T^{8} + 150285932 p T^{9} + 17627940 p^{2} T^{10} + 1830390 p^{3} T^{11} + 167990 p^{4} T^{12} + 12938 p^{5} T^{13} + 812 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 10 T + 220 T^{2} - 1858 T^{3} + 23988 T^{4} - 172914 T^{5} + 1835696 T^{6} - 12425750 T^{7} + 115955634 T^{8} - 12425750 p T^{9} + 1835696 p^{2} T^{10} - 172914 p^{3} T^{11} + 23988 p^{4} T^{12} - 1858 p^{5} T^{13} + 220 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 22 T + 494 T^{2} + 7064 T^{3} + 98100 T^{4} + 1063800 T^{5} + 11217818 T^{6} + 98161050 T^{7} + 834720538 T^{8} + 98161050 p T^{9} + 11217818 p^{2} T^{10} + 1063800 p^{3} T^{11} + 98100 p^{4} T^{12} + 7064 p^{5} T^{13} + 494 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 14 T + 476 T^{2} + 4826 T^{3} + 96208 T^{4} + 779224 T^{5} + 11672804 T^{6} + 78404804 T^{7} + 946709457 T^{8} + 78404804 p T^{9} + 11672804 p^{2} T^{10} + 779224 p^{3} T^{11} + 96208 p^{4} T^{12} + 4826 p^{5} T^{13} + 476 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 10 T + 268 T^{2} + 2392 T^{3} + 43712 T^{4} + 335868 T^{5} + 4736008 T^{6} + 32269814 T^{7} + 387499578 T^{8} + 32269814 p T^{9} + 4736008 p^{2} T^{10} + 335868 p^{3} T^{11} + 43712 p^{4} T^{12} + 2392 p^{5} T^{13} + 268 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 12 T + 422 T^{2} + 4186 T^{3} + 79840 T^{4} + 674098 T^{5} + 9324518 T^{6} + 68642544 T^{7} + 781969530 T^{8} + 68642544 p T^{9} + 9324518 p^{2} T^{10} + 674098 p^{3} T^{11} + 79840 p^{4} T^{12} + 4186 p^{5} T^{13} + 422 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 364 T^{2} - 4058 T^{3} + 64540 T^{4} - 617566 T^{5} + 7900400 T^{6} - 65652218 T^{7} + 717456169 T^{8} - 65652218 p T^{9} + 7900400 p^{2} T^{10} - 617566 p^{3} T^{11} + 64540 p^{4} T^{12} - 4058 p^{5} T^{13} + 364 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 24 T + 608 T^{2} - 7784 T^{3} + 102700 T^{4} - 702968 T^{5} + 5553504 T^{6} - 4107496 T^{7} + 108377478 T^{8} - 4107496 p T^{9} + 5553504 p^{2} T^{10} - 702968 p^{3} T^{11} + 102700 p^{4} T^{12} - 7784 p^{5} T^{13} + 608 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 2 T + 308 T^{2} - 1250 T^{3} + 56308 T^{4} - 270150 T^{5} + 7434748 T^{6} - 34187158 T^{7} + 760540582 T^{8} - 34187158 p T^{9} + 7434748 p^{2} T^{10} - 270150 p^{3} T^{11} + 56308 p^{4} T^{12} - 1250 p^{5} T^{13} + 308 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 16 T + 558 T^{2} - 7254 T^{3} + 143022 T^{4} - 1535640 T^{5} + 22814312 T^{6} - 206722248 T^{7} + 2575665135 T^{8} - 206722248 p T^{9} + 22814312 p^{2} T^{10} - 1535640 p^{3} T^{11} + 143022 p^{4} T^{12} - 7254 p^{5} T^{13} + 558 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.79069735440416142096734968067, −3.67757173895337074152602344379, −3.43387545137454471577211557489, −3.42036901506473033558093761509, −3.40339410486236697175183003401, −3.31699555329675008491935562898, −3.27651904085541748209327891295, −3.19138098496613562075506192956, −2.83756041712934348850937437767, −2.56164899564444306440951606120, −2.47797257986356344889541342258, −2.28954249338576868779497268725, −2.25160620738934515115529457274, −2.24186902590545908027434529950, −2.10188754859870632565409448429, −1.72576378066734052531245014187, −1.70296721784786368682892325870, −1.69175620136474834163604787866, −1.43085575208636500748934368775, −1.37134446890778160572305394853, −1.14306820612538573238529895191, −1.13839327254742910999212539798, −1.06607908178000666486128479813, −0.988529754832901018149703621719, −0.890257089635860173340848916331, 0, 0, 0, 0, 0, 0, 0, 0, 0.890257089635860173340848916331, 0.988529754832901018149703621719, 1.06607908178000666486128479813, 1.13839327254742910999212539798, 1.14306820612538573238529895191, 1.37134446890778160572305394853, 1.43085575208636500748934368775, 1.69175620136474834163604787866, 1.70296721784786368682892325870, 1.72576378066734052531245014187, 2.10188754859870632565409448429, 2.24186902590545908027434529950, 2.25160620738934515115529457274, 2.28954249338576868779497268725, 2.47797257986356344889541342258, 2.56164899564444306440951606120, 2.83756041712934348850937437767, 3.19138098496613562075506192956, 3.27651904085541748209327891295, 3.31699555329675008491935562898, 3.40339410486236697175183003401, 3.42036901506473033558093761509, 3.43387545137454471577211557489, 3.67757173895337074152602344379, 3.79069735440416142096734968067
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.