Dirichlet series
| L(s) = 1 | − 48·17-s − 96·19-s − 8·23-s − 68·25-s − 80·29-s + 48·31-s − 64·37-s − 112·43-s + 264·47-s − 72·53-s − 168·59-s − 144·61-s + 32·67-s − 224·71-s + 336·73-s + 216·79-s − 96·89-s − 24·101-s − 96·103-s + 328·107-s − 8·109-s − 512·113-s + 360·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.82·17-s − 5.05·19-s − 0.347·23-s − 2.71·25-s − 2.75·29-s + 1.54·31-s − 1.72·37-s − 2.60·43-s + 5.61·47-s − 1.35·53-s − 2.84·59-s − 2.36·61-s + 0.477·67-s − 3.15·71-s + 4.60·73-s + 2.73·79-s − 1.07·89-s − 0.237·101-s − 0.932·103-s + 3.06·107-s − 0.0733·109-s − 4.53·113-s + 2.97·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.84884\times 10^{13}\) |
| Root analytic conductor: | \(6.93293\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.5143721969\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5143721969\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 68 T^{2} + 492 p T^{4} - 1032 T^{5} + 62344 T^{6} - 51936 T^{7} + 1407071 T^{8} - 51936 p^{2} T^{9} + 62344 p^{4} T^{10} - 1032 p^{6} T^{11} + 492 p^{9} T^{12} + 68 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 360 T^{2} + 96 T^{3} + 68470 T^{4} - 23088 T^{5} - 11462976 T^{6} + 1247520 T^{7} + 1624765635 T^{8} + 1247520 p^{2} T^{9} - 11462976 p^{4} T^{10} - 23088 p^{6} T^{11} + 68470 p^{8} T^{12} + 96 p^{10} T^{13} - 360 p^{12} T^{14} + p^{16} T^{16} \) | |
| 13 | \( 1 - 640 T^{2} + 201024 T^{4} - 3606656 p T^{6} + 8914586690 T^{8} - 3606656 p^{5} T^{10} + 201024 p^{8} T^{12} - 640 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 + 48 T + 1924 T^{2} + 192 p^{2} T^{3} + 1429948 T^{4} + 109080 p^{2} T^{5} + 643753096 T^{6} + 11956044720 T^{7} + 209982732223 T^{8} + 11956044720 p^{2} T^{9} + 643753096 p^{4} T^{10} + 109080 p^{8} T^{11} + 1429948 p^{8} T^{12} + 192 p^{12} T^{13} + 1924 p^{12} T^{14} + 48 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 96 T + 5484 T^{2} + 231552 T^{3} + 8024850 T^{4} + 237456576 T^{5} + 6114429648 T^{6} + 138708232608 T^{7} + 2791933387859 T^{8} + 138708232608 p^{2} T^{9} + 6114429648 p^{4} T^{10} + 237456576 p^{6} T^{11} + 8024850 p^{8} T^{12} + 231552 p^{10} T^{13} + 5484 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 8 T - 1440 T^{2} + 6352 T^{3} + 1245734 T^{4} - 10123272 T^{5} - 656508800 T^{6} + 2853951224 T^{7} + 290531534499 T^{8} + 2853951224 p^{2} T^{9} - 656508800 p^{4} T^{10} - 10123272 p^{6} T^{11} + 1245734 p^{8} T^{12} + 6352 p^{10} T^{13} - 1440 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( ( 1 + 40 T + 1440 T^{2} + 35480 T^{3} + 1476410 T^{4} + 35480 p^{2} T^{5} + 1440 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 - 48 T + 1772 T^{2} - 48192 T^{3} + 547506 T^{4} + 1137264 T^{5} - 1177108784 T^{6} + 88094121936 T^{7} - 2921554099981 T^{8} + 88094121936 p^{2} T^{9} - 1177108784 p^{4} T^{10} + 1137264 p^{6} T^{11} + 547506 p^{8} T^{12} - 48192 p^{10} T^{13} + 1772 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 64 T + 2112 T^{2} + 59648 T^{3} + 644126 T^{4} - 32741760 T^{5} + 3363364864 T^{6} + 257607956800 T^{7} + 9871545282723 T^{8} + 257607956800 p^{2} T^{9} + 3363364864 p^{4} T^{10} - 32741760 p^{6} T^{11} + 644126 p^{8} T^{12} + 59648 p^{10} T^{13} + 2112 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 3640 T^{2} + 14914392 T^{4} - 31945062104 T^{6} + 68419019701298 T^{8} - 31945062104 p^{4} T^{10} + 14914392 p^{8} T^{12} - 3640 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( ( 1 + 56 T + 84 p T^{2} + 107464 T^{3} + 6712358 T^{4} + 107464 p^{2} T^{5} + 84 p^{5} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 47 | \( 1 - 264 T + 37940 T^{2} - 3882912 T^{3} + 314740146 T^{4} - 21398188008 T^{5} + 1268985966640 T^{6} - 67642086086760 T^{7} + 3306937422359987 T^{8} - 67642086086760 p^{2} T^{9} + 1268985966640 p^{4} T^{10} - 21398188008 p^{6} T^{11} + 314740146 p^{8} T^{12} - 3882912 p^{10} T^{13} + 37940 p^{12} T^{14} - 264 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 72 T - 588 T^{2} + 178032 T^{3} + 16582090 T^{4} - 158994312 T^{5} + 33946611408 T^{6} + 1995258747768 T^{7} - 68021885502093 T^{8} + 1995258747768 p^{2} T^{9} + 33946611408 p^{4} T^{10} - 158994312 p^{6} T^{11} + 16582090 p^{8} T^{12} + 178032 p^{10} T^{13} - 588 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 168 T + 16692 T^{2} + 1223712 T^{3} + 60611538 T^{4} + 1000432392 T^{5} - 152463705552 T^{6} - 20026557780792 T^{7} - 1451602798653997 T^{8} - 20026557780792 p^{2} T^{9} - 152463705552 p^{4} T^{10} + 1000432392 p^{6} T^{11} + 60611538 p^{8} T^{12} + 1223712 p^{10} T^{13} + 16692 p^{12} T^{14} + 168 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 144 T + 18656 T^{2} + 1691136 T^{3} + 153918528 T^{4} + 12653747232 T^{5} + 923469869632 T^{6} + 63336115849200 T^{7} + 3836802319781471 T^{8} + 63336115849200 p^{2} T^{9} + 923469869632 p^{4} T^{10} + 12653747232 p^{6} T^{11} + 153918528 p^{8} T^{12} + 1691136 p^{10} T^{13} + 18656 p^{12} T^{14} + 144 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 32 T - 10884 T^{2} - 48064 T^{3} + 70418666 T^{4} + 1497416352 T^{5} - 302846952464 T^{6} - 3770996900576 T^{7} + 1160434720244979 T^{8} - 3770996900576 p^{2} T^{9} - 302846952464 p^{4} T^{10} + 1497416352 p^{6} T^{11} + 70418666 p^{8} T^{12} - 48064 p^{10} T^{13} - 10884 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 112 T + 13704 T^{2} + 1059392 T^{3} + 79617770 T^{4} + 1059392 p^{2} T^{5} + 13704 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 336 T + 66528 T^{2} - 9709056 T^{3} + 1185969600 T^{4} - 126302528832 T^{5} + 11936480262720 T^{6} - 1009238088017328 T^{7} + 77237278241369279 T^{8} - 1009238088017328 p^{2} T^{9} + 11936480262720 p^{4} T^{10} - 126302528832 p^{6} T^{11} + 1185969600 p^{8} T^{12} - 9709056 p^{10} T^{13} + 66528 p^{12} T^{14} - 336 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 216 T + 9252 T^{2} + 133680 T^{3} + 130134922 T^{4} - 180187224 p T^{5} - 44437168368 T^{6} - 18348496114728 T^{7} + 7297444984354515 T^{8} - 18348496114728 p^{2} T^{9} - 44437168368 p^{4} T^{10} - 180187224 p^{7} T^{11} + 130134922 p^{8} T^{12} + 133680 p^{10} T^{13} + 9252 p^{12} T^{14} - 216 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 35400 T^{2} + 603816348 T^{4} - 6617387785848 T^{6} + 52566528325929350 T^{8} - 6617387785848 p^{4} T^{10} + 603816348 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} \) | |
| 89 | \( 1 + 96 T + 15492 T^{2} + 1192320 T^{3} + 93673308 T^{4} - 4376375256 T^{5} - 630708975480 T^{6} - 151171483500672 T^{7} - 13302455367766657 T^{8} - 151171483500672 p^{2} T^{9} - 630708975480 p^{4} T^{10} - 4376375256 p^{6} T^{11} + 93673308 p^{8} T^{12} + 1192320 p^{10} T^{13} + 15492 p^{12} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 61632 T^{2} + 1744804416 T^{4} - 29860121715648 T^{6} + 339981119551810562 T^{8} - 29860121715648 p^{4} T^{10} + 1744804416 p^{8} T^{12} - 61632 p^{12} T^{14} + p^{16} T^{16} \) | |
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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
−3.80720299446141687271274391290, −3.72211045234780522419539589220, −3.42255923543732305447884947690, −3.38792774218030366088526732419, −3.27261482367572828328189716885, −3.25313650428209334624663794733, −2.80413418764312222246839495699, −2.66726070221074051700445180966, −2.53838851101818178671711980980, −2.42594691879907451591520146914, −2.39028397700790036406175191752, −2.30887256003744516884899628342, −2.14181469817116752872774305573, −2.06237681351871475703748067688, −1.80420375802518089175639871628, −1.77606015377244624829886917859, −1.61746667838493583680456422307, −1.55359913307349186576507165125, −1.28704228395602373494050904377, −1.20372187565671766066202313094, −0.67410453183815055571797847099, −0.51422667320134399295908825352, −0.26248340041770569345538711694, −0.25603272363798097412302059293, −0.14569096854612154654395077733, 0.14569096854612154654395077733, 0.25603272363798097412302059293, 0.26248340041770569345538711694, 0.51422667320134399295908825352, 0.67410453183815055571797847099, 1.20372187565671766066202313094, 1.28704228395602373494050904377, 1.55359913307349186576507165125, 1.61746667838493583680456422307, 1.77606015377244624829886917859, 1.80420375802518089175639871628, 2.06237681351871475703748067688, 2.14181469817116752872774305573, 2.30887256003744516884899628342, 2.39028397700790036406175191752, 2.42594691879907451591520146914, 2.53838851101818178671711980980, 2.66726070221074051700445180966, 2.80413418764312222246839495699, 3.25313650428209334624663794733, 3.27261482367572828328189716885, 3.38792774218030366088526732419, 3.42255923543732305447884947690, 3.72211045234780522419539589220, 3.80720299446141687271274391290
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.