Dirichlet series
| L(s) = 1 | + 4-s + 10·5-s + 4·11-s − 10·17-s + 10·19-s + 10·20-s − 10·23-s + 55·25-s − 10·29-s + 6·31-s − 10·37-s + 14·41-s + 4·44-s + 30·47-s + 20·49-s + 40·55-s − 14·61-s + 10·67-s − 10·68-s + 34·71-s + 10·76-s − 50·83-s − 100·85-s − 20·89-s − 10·92-s + 100·95-s − 20·97-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 4.47·5-s + 1.20·11-s − 2.42·17-s + 2.29·19-s + 2.23·20-s − 2.08·23-s + 11·25-s − 1.85·29-s + 1.07·31-s − 1.64·37-s + 2.18·41-s + 0.603·44-s + 4.37·47-s + 20/7·49-s + 5.39·55-s − 1.79·61-s + 1.22·67-s − 1.21·68-s + 4.03·71-s + 1.14·76-s − 5.48·83-s − 10.8·85-s − 2.11·89-s − 1.04·92-s + 10.2·95-s − 2.03·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(27791.8\) |
| Root analytic conductor: | \(1.89559\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(18.92195037\) |
| \(L(\frac12)\) | \(\approx\) | \(18.92195037\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 2 p T + 9 p T^{2} - 26 p T^{3} + 61 p T^{4} - 26 p^{2} T^{5} + 9 p^{3} T^{6} - 2 p^{4} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( 1 - 20 T^{2} + 146 T^{4} - 160 T^{6} - 2789 T^{8} - 160 p^{2} T^{10} + 146 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 4 T - 25 T^{2} + 80 T^{3} + 260 T^{4} - 752 T^{5} + 123 p T^{6} + 210 p T^{7} - 44765 T^{8} + 210 p^{2} T^{9} + 123 p^{3} T^{10} - 752 p^{3} T^{11} + 260 p^{4} T^{12} + 80 p^{5} T^{13} - 25 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T^{2} + 20 T^{3} + 96 T^{4} - 100 T^{5} + 25 T^{6} + 6470 T^{7} - 6509 T^{8} + 6470 p T^{9} + 25 p^{2} T^{10} - 100 p^{3} T^{11} + 96 p^{4} T^{12} + 20 p^{5} T^{13} - 5 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 10 T + 75 T^{2} + 420 T^{3} + 2171 T^{4} + 9960 T^{5} + 43585 T^{6} + 177290 T^{7} + 711536 T^{8} + 177290 p T^{9} + 43585 p^{2} T^{10} + 9960 p^{3} T^{11} + 2171 p^{4} T^{12} + 420 p^{5} T^{13} + 75 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 10 T + 22 T^{2} - 30 T^{3} + 1023 T^{4} - 4210 T^{5} - 4736 T^{6} - 17200 T^{7} + 424105 T^{8} - 17200 p T^{9} - 4736 p^{2} T^{10} - 4210 p^{3} T^{11} + 1023 p^{4} T^{12} - 30 p^{5} T^{13} + 22 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 10 T + 105 T^{2} + 790 T^{3} + 5156 T^{4} + 30260 T^{5} + 156735 T^{6} + 798320 T^{7} + 3801671 T^{8} + 798320 p T^{9} + 156735 p^{2} T^{10} + 30260 p^{3} T^{11} + 5156 p^{4} T^{12} + 790 p^{5} T^{13} + 105 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 10 T + 47 T^{2} + 470 T^{3} + 4888 T^{4} + 28460 T^{5} + 139689 T^{6} + 999650 T^{7} + 6593475 T^{8} + 999650 p T^{9} + 139689 p^{2} T^{10} + 28460 p^{3} T^{11} + 4888 p^{4} T^{12} + 470 p^{5} T^{13} + 47 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 113520 p T^{9} - 9417 p^{2} T^{10} - 4048 p^{3} T^{11} + 1840 p^{4} T^{12} - 110 p^{5} T^{13} + 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 10 T + 65 T^{2} + 230 T^{3} + 1471 T^{4} + 9410 T^{5} + 48855 T^{6} - 117890 T^{7} - 967744 T^{8} - 117890 p T^{9} + 48855 p^{2} T^{10} + 9410 p^{3} T^{11} + 1471 p^{4} T^{12} + 230 p^{5} T^{13} + 65 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 14 T + 45 T^{2} - 130 T^{3} + 2680 T^{4} - 7652 T^{5} - 46977 T^{6} + 402900 T^{7} - 2944785 T^{8} + 402900 p T^{9} - 46977 p^{2} T^{10} - 7652 p^{3} T^{11} + 2680 p^{4} T^{12} - 130 p^{5} T^{13} + 45 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 120 T^{2} + 10666 T^{4} - 14800 p T^{6} + 31214171 T^{8} - 14800 p^{3} T^{10} + 10666 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 30 T + 425 T^{2} - 3450 T^{3} + 13416 T^{4} + 40020 T^{5} - 992525 T^{6} + 8396100 T^{7} - 57415609 T^{8} + 8396100 p T^{9} - 992525 p^{2} T^{10} + 40020 p^{3} T^{11} + 13416 p^{4} T^{12} - 3450 p^{5} T^{13} + 425 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 90 T^{2} - 900 T^{3} + 5291 T^{4} - 81000 T^{5} + 628380 T^{6} - 3853800 T^{7} + 44701381 T^{8} - 3853800 p T^{9} + 628380 p^{2} T^{10} - 81000 p^{3} T^{11} + 5291 p^{4} T^{12} - 900 p^{5} T^{13} + 90 p^{6} T^{14} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 p T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 324004 p^{2} T^{10} + 3563 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 1508020 p T^{9} - 186192 p^{2} T^{10} + 25262 p^{3} T^{11} + 5215 p^{4} T^{12} - 30 p^{5} T^{13} + 30 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 10 T + 195 T^{2} - 2790 T^{3} + 29716 T^{4} - 356960 T^{5} + 3452465 T^{6} - 30345750 T^{7} + 284471411 T^{8} - 30345750 p T^{9} + 3452465 p^{2} T^{10} - 356960 p^{3} T^{11} + 29716 p^{4} T^{12} - 2790 p^{5} T^{13} + 195 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 34 T + 425 T^{2} - 2570 T^{3} + 16960 T^{4} - 219512 T^{5} + 2212323 T^{6} - 19311440 T^{7} + 171308535 T^{8} - 19311440 p T^{9} + 2212323 p^{2} T^{10} - 219512 p^{3} T^{11} + 16960 p^{4} T^{12} - 2570 p^{5} T^{13} + 425 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 110 T^{2} + 840 T^{3} + 6291 T^{4} + 92400 T^{5} + 907660 T^{6} + 5561520 T^{7} + 91719461 T^{8} + 5561520 p T^{9} + 907660 p^{2} T^{10} + 92400 p^{3} T^{11} + 6291 p^{4} T^{12} + 840 p^{5} T^{13} + 110 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 - 138 T^{2} + 1500 T^{3} + 4403 T^{4} - 205200 T^{5} + 1430844 T^{6} + 7779600 T^{7} - 206126795 T^{8} + 7779600 p T^{9} + 1430844 p^{2} T^{10} - 205200 p^{3} T^{11} + 4403 p^{4} T^{12} + 1500 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 + 50 T + 1335 T^{2} + 23750 T^{3} + 305836 T^{4} + 2886800 T^{5} + 19447445 T^{6} + 89310250 T^{7} + 442867371 T^{8} + 89310250 p T^{9} + 19447445 p^{2} T^{10} + 2886800 p^{3} T^{11} + 305836 p^{4} T^{12} + 23750 p^{5} T^{13} + 1335 p^{6} T^{14} + 50 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 1290 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 20 T + 270 T^{2} + 3780 T^{3} + 51031 T^{4} + 591220 T^{5} + 6636980 T^{6} + 69655800 T^{7} + 673071701 T^{8} + 69655800 p T^{9} + 6636980 p^{2} T^{10} + 591220 p^{3} T^{11} + 51031 p^{4} T^{12} + 3780 p^{5} T^{13} + 270 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.07620497485100582634757179740, −4.96146235561373259170084137753, −4.38789970232017488202271127581, −4.34598396102670409490877609647, −4.29289680855928684608547287651, −4.21471931981816256278024760264, −4.13352061325923610258425166067, −4.09264399318219194788688131903, −3.85968454534324081402546219843, −3.37613307584424556915080177567, −3.31115022557293392151602151661, −3.29640215641836863737989102656, −2.77094965907302117364652784623, −2.74194146982938034362459465295, −2.67344976533614005362310430609, −2.51249355014333875321363756653, −2.39984390288848025436439012250, −2.05373170810165585226668172506, −1.90607243239857289056459172843, −1.89566655155295930216894338322, −1.78123874102214784031766967759, −1.41448181658267477941922708751, −1.19601230632010696983992745145, −0.869599536643534136925069399850, −0.66808173566470533336057162955, 0.66808173566470533336057162955, 0.869599536643534136925069399850, 1.19601230632010696983992745145, 1.41448181658267477941922708751, 1.78123874102214784031766967759, 1.89566655155295930216894338322, 1.90607243239857289056459172843, 2.05373170810165585226668172506, 2.39984390288848025436439012250, 2.51249355014333875321363756653, 2.67344976533614005362310430609, 2.74194146982938034362459465295, 2.77094965907302117364652784623, 3.29640215641836863737989102656, 3.31115022557293392151602151661, 3.37613307584424556915080177567, 3.85968454534324081402546219843, 4.09264399318219194788688131903, 4.13352061325923610258425166067, 4.21471931981816256278024760264, 4.29289680855928684608547287651, 4.34598396102670409490877609647, 4.38789970232017488202271127581, 4.96146235561373259170084137753, 5.07620497485100582634757179740
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.