Dirichlet series
| L(s) = 1 | + 8·3-s − 8·5-s + 8·7-s + 28·9-s + 16·11-s − 16·13-s − 64·15-s − 16·17-s + 64·21-s + 16·23-s + 40·25-s + 56·27-s − 8·29-s + 8·31-s + 128·33-s − 64·35-s − 40·37-s − 128·39-s + 32·43-s − 224·45-s + 16·47-s + 28·49-s − 128·51-s + 16·53-s − 128·55-s + 8·59-s + 24·61-s + ⋯ |
| L(s) = 1 | + 4.61·3-s − 3.57·5-s + 3.02·7-s + 28/3·9-s + 4.82·11-s − 4.43·13-s − 16.5·15-s − 3.88·17-s + 13.9·21-s + 3.33·23-s + 8·25-s + 10.7·27-s − 1.48·29-s + 1.43·31-s + 22.2·33-s − 10.8·35-s − 6.57·37-s − 20.4·39-s + 4.87·43-s − 33.3·45-s + 2.33·47-s + 4·49-s − 17.9·51-s + 2.19·53-s − 17.2·55-s + 1.04·59-s + 3.07·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{40} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(126767.\) |
| Root analytic conductor: | \(2.08419\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{40} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.72198227\) |
| \(L(\frac12)\) | \(\approx\) | \(23.72198227\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 16 T + 144 T^{2} + 880 T^{3} + 4130 T^{4} + 880 p T^{5} + 144 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - 8 T + 4 p^{2} T^{2} - 40 p T^{3} + 316 T^{4} - 688 T^{5} + 1316 T^{6} - 2320 T^{7} + 3992 T^{8} - 2320 p T^{9} + 1316 p^{2} T^{10} - 688 p^{3} T^{11} + 316 p^{4} T^{12} - 40 p^{6} T^{13} + 4 p^{8} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 8 T + 24 T^{2} + 24 T^{3} + 18 T^{4} + 296 T^{5} + 1096 T^{6} + 1144 T^{7} - 158 T^{8} + 1144 p T^{9} + 1096 p^{2} T^{10} + 296 p^{3} T^{11} + 18 p^{4} T^{12} + 24 p^{5} T^{13} + 24 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 8 T + 36 T^{2} - 152 T^{3} + 508 T^{4} - 1328 T^{5} + 3396 T^{6} - 8016 T^{7} + 18712 T^{8} - 8016 p T^{9} + 3396 p^{2} T^{10} - 1328 p^{3} T^{11} + 508 p^{4} T^{12} - 152 p^{5} T^{13} + 36 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 16 T + 100 T^{2} - 24 p T^{3} + 84 T^{4} + 576 T^{5} - 1252 T^{6} + 27080 T^{7} - 156616 T^{8} + 27080 p T^{9} - 1252 p^{2} T^{10} + 576 p^{3} T^{11} + 84 p^{4} T^{12} - 24 p^{6} T^{13} + 100 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 16 T + 128 T^{2} + 768 T^{3} + 3968 T^{4} + 17792 T^{5} + 71680 T^{6} + 273360 T^{7} + 1004194 T^{8} + 273360 p T^{9} + 71680 p^{2} T^{10} + 17792 p^{3} T^{11} + 3968 p^{4} T^{12} + 768 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 32 T^{2} - 80 T^{3} + 512 T^{4} + 2832 T^{5} - 1184 T^{6} - 22528 T^{7} - 100062 T^{8} - 22528 p T^{9} - 1184 p^{2} T^{10} + 2832 p^{3} T^{11} + 512 p^{4} T^{12} - 80 p^{5} T^{13} - 32 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 - 16 T + 100 T^{2} - 168 T^{3} - 876 T^{4} + 3456 T^{5} - 7492 T^{6} + 197672 T^{7} - 1570888 T^{8} + 197672 p T^{9} - 7492 p^{2} T^{10} + 3456 p^{3} T^{11} - 876 p^{4} T^{12} - 168 p^{5} T^{13} + 100 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 8 T - 20 T^{2} - 552 T^{3} - 1146 T^{4} + 20760 T^{5} + 125900 T^{6} - 227320 T^{7} - 4644478 T^{8} - 227320 p T^{9} + 125900 p^{2} T^{10} + 20760 p^{3} T^{11} - 1146 p^{4} T^{12} - 552 p^{5} T^{13} - 20 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 8 T + 92 T^{2} - 320 T^{3} + 3604 T^{4} - 392 p T^{5} + 155636 T^{6} - 13536 p T^{7} + 157064 p T^{8} - 13536 p^{2} T^{9} + 155636 p^{2} T^{10} - 392 p^{4} T^{11} + 3604 p^{4} T^{12} - 320 p^{5} T^{13} + 92 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 40 T + 696 T^{2} + 6520 T^{3} + 29874 T^{4} - 14456 T^{5} - 995480 T^{6} - 5900904 T^{7} - 27444062 T^{8} - 5900904 p T^{9} - 995480 p^{2} T^{10} - 14456 p^{3} T^{11} + 29874 p^{4} T^{12} + 6520 p^{5} T^{13} + 696 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 44 T^{2} + 512 T^{3} + 934 T^{4} - 28416 T^{5} + 158132 T^{6} + 663808 T^{7} - 10407486 T^{8} + 663808 p T^{9} + 158132 p^{2} T^{10} - 28416 p^{3} T^{11} + 934 p^{4} T^{12} + 512 p^{5} T^{13} - 44 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 - 32 T + 432 T^{2} - 2736 T^{3} + 640 T^{4} + 122992 T^{5} - 869968 T^{6} + 1577248 T^{7} + 6587042 T^{8} + 1577248 p T^{9} - 869968 p^{2} T^{10} + 122992 p^{3} T^{11} + 640 p^{4} T^{12} - 2736 p^{5} T^{13} + 432 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 8468 T^{4} + 1232 T^{5} - 134784 T^{6} + 2691504 T^{7} - 33833882 T^{8} + 2691504 p T^{9} - 134784 p^{2} T^{10} + 1232 p^{3} T^{11} + 8468 p^{4} T^{12} - 1392 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 16 T + 196 T^{2} - 2224 T^{3} + 21000 T^{4} - 179216 T^{5} + 1498924 T^{6} - 11528112 T^{7} + 83946542 T^{8} - 11528112 p T^{9} + 1498924 p^{2} T^{10} - 179216 p^{3} T^{11} + 21000 p^{4} T^{12} - 2224 p^{5} T^{13} + 196 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 8 T + 40 T^{2} + 664 T^{3} - 6048 T^{4} + 36616 T^{5} + 180968 T^{6} - 1710680 T^{7} + 25804610 T^{8} - 1710680 p T^{9} + 180968 p^{2} T^{10} + 36616 p^{3} T^{11} - 6048 p^{4} T^{12} + 664 p^{5} T^{13} + 40 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 24 T + 376 T^{2} - 4344 T^{3} + 42450 T^{4} - 372936 T^{5} + 3081640 T^{6} - 24675240 T^{7} + 191910626 T^{8} - 24675240 p T^{9} + 3081640 p^{2} T^{10} - 372936 p^{3} T^{11} + 42450 p^{4} T^{12} - 4344 p^{5} T^{13} + 376 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 16 T + 196 T^{2} - 1040 T^{3} + 8590 T^{4} - 1040 p T^{5} + 196 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 8 T + 52 T^{2} - 648 T^{3} + 1404 T^{4} + 2976 T^{5} - 342076 T^{6} + 5998144 T^{7} - 41219752 T^{8} + 5998144 p T^{9} - 342076 p^{2} T^{10} + 2976 p^{3} T^{11} + 1404 p^{4} T^{12} - 648 p^{5} T^{13} + 52 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 24 T + 364 T^{2} - 5736 T^{3} + 73030 T^{4} - 782216 T^{5} + 8485068 T^{6} - 81253240 T^{7} + 689934338 T^{8} - 81253240 p T^{9} + 8485068 p^{2} T^{10} - 782216 p^{3} T^{11} + 73030 p^{4} T^{12} - 5736 p^{5} T^{13} + 364 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 284 T^{2} - 3248 T^{3} + 31956 T^{4} - 257656 T^{5} + 2982580 T^{6} - 31942272 T^{7} + 273528312 T^{8} - 31942272 p T^{9} + 2982580 p^{2} T^{10} - 257656 p^{3} T^{11} + 31956 p^{4} T^{12} - 3248 p^{5} T^{13} + 284 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 8 T + 216 T^{2} + 2184 T^{3} + 28000 T^{4} + 310488 T^{5} + 3001496 T^{6} + 30916184 T^{7} + 272377922 T^{8} + 30916184 p T^{9} + 3001496 p^{2} T^{10} + 310488 p^{3} T^{11} + 28000 p^{4} T^{12} + 2184 p^{5} T^{13} + 216 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 32 T + 512 T^{2} + 7504 T^{3} + 116512 T^{4} + 1475408 T^{5} + 15713920 T^{6} + 170899872 T^{7} + 1762186818 T^{8} + 170899872 p T^{9} + 15713920 p^{2} T^{10} + 1475408 p^{3} T^{11} + 116512 p^{4} T^{12} + 7504 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 8 T + 168 T^{2} + 1112 T^{3} + 3346 T^{4} - 46936 T^{5} - 1062312 T^{6} - 23230920 T^{7} - 143196062 T^{8} - 23230920 p T^{9} - 1062312 p^{2} T^{10} - 46936 p^{3} T^{11} + 3346 p^{4} T^{12} + 1112 p^{5} T^{13} + 168 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.67358922343948907025009486671, −4.35281595146888677325317785615, −4.21754852745345670915245592640, −4.18145662903227874154376905352, −4.09043546589682039863128180201, −3.97027404362845244749054616603, −3.86918162936026080113508670178, −3.70707682029600685814412048174, −3.70156169547523327411226080950, −3.54069753092911511345386914528, −3.39459147127658861945561480381, −3.13515769475501539860400255555, −2.90371683601490841634746373523, −2.72361530438564849836435117508, −2.52353638571951002244979194039, −2.39719006468243916682251334783, −2.35319757614790467190654464166, −2.25483238750895433165798915692, −2.21015569466994424442300171148, −1.96758724460682588918050020440, −1.63229664846240747359049031519, −1.16605083945495677340367738124, −1.16424913415070108348503547969, −0.73672851997683773907495063667, −0.55564628205413507078714024357, 0.55564628205413507078714024357, 0.73672851997683773907495063667, 1.16424913415070108348503547969, 1.16605083945495677340367738124, 1.63229664846240747359049031519, 1.96758724460682588918050020440, 2.21015569466994424442300171148, 2.25483238750895433165798915692, 2.35319757614790467190654464166, 2.39719006468243916682251334783, 2.52353638571951002244979194039, 2.72361530438564849836435117508, 2.90371683601490841634746373523, 3.13515769475501539860400255555, 3.39459147127658861945561480381, 3.54069753092911511345386914528, 3.70156169547523327411226080950, 3.70707682029600685814412048174, 3.86918162936026080113508670178, 3.97027404362845244749054616603, 4.09043546589682039863128180201, 4.18145662903227874154376905352, 4.21754852745345670915245592640, 4.35281595146888677325317785615, 4.67358922343948907025009486671
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.