Dirichlet series
| L(s) = 1 | − 2·2-s − 8·3-s + 16·6-s + 2·7-s + 3·8-s + 36·9-s + 6·11-s + 6·13-s − 4·14-s − 12·17-s − 72·18-s − 16·21-s − 12·22-s − 14·23-s − 24·24-s − 12·26-s − 120·27-s + 8·29-s + 8·31-s − 4·32-s − 48·33-s + 24·34-s + 4·37-s − 48·39-s + 2·41-s + 32·42-s + 2·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.61·3-s + 6.53·6-s + 0.755·7-s + 1.06·8-s + 12·9-s + 1.80·11-s + 1.66·13-s − 1.06·14-s − 2.91·17-s − 16.9·18-s − 3.49·21-s − 2.55·22-s − 2.91·23-s − 4.89·24-s − 2.35·26-s − 23.0·27-s + 1.48·29-s + 1.43·31-s − 0.707·32-s − 8.35·33-s + 4.11·34-s + 0.657·37-s − 7.68·39-s + 0.312·41-s + 4.93·42-s + 0.304·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{16} \cdot 29^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.27734\times 10^{9}\) |
| Root analytic conductor: | \(4.16742\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{16} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.026294964\) |
| \(L(\frac12)\) | \(\approx\) | \(1.026294964\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{8} \) |
| 5 | \( 1 \) | |
| 29 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( 1 + p T + p^{2} T^{2} + 5 T^{3} + p^{2} T^{4} + T^{6} - p T^{7} + 3 p T^{8} - p^{2} T^{9} + p^{2} T^{10} + p^{6} T^{12} + 5 p^{5} T^{13} + p^{8} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 2 T + 11 T^{2} - 54 T^{3} + 149 T^{4} - 248 T^{5} + 1381 T^{6} - 2678 T^{7} + 4659 T^{8} - 2678 p T^{9} + 1381 p^{2} T^{10} - 248 p^{3} T^{11} + 149 p^{4} T^{12} - 54 p^{5} T^{13} + 11 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 6 T + 52 T^{2} - 224 T^{3} + 1277 T^{4} - 4538 T^{5} + 20835 T^{6} - 64084 T^{7} + 257338 T^{8} - 64084 p T^{9} + 20835 p^{2} T^{10} - 4538 p^{3} T^{11} + 1277 p^{4} T^{12} - 224 p^{5} T^{13} + 52 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 6 T + 43 T^{2} - 136 T^{3} + 837 T^{4} - 2474 T^{5} + 15019 T^{6} - 42580 T^{7} + 228905 T^{8} - 42580 p T^{9} + 15019 p^{2} T^{10} - 2474 p^{3} T^{11} + 837 p^{4} T^{12} - 136 p^{5} T^{13} + 43 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 12 T + 126 T^{2} + 832 T^{3} + 5153 T^{4} + 25034 T^{5} + 123885 T^{6} + 523636 T^{7} + 2323644 T^{8} + 523636 p T^{9} + 123885 p^{2} T^{10} + 25034 p^{3} T^{11} + 5153 p^{4} T^{12} + 832 p^{5} T^{13} + 126 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 53 T^{2} - 64 T^{3} + 2034 T^{4} - 3912 T^{5} + 49091 T^{6} - 125480 T^{7} + 1069082 T^{8} - 125480 p T^{9} + 49091 p^{2} T^{10} - 3912 p^{3} T^{11} + 2034 p^{4} T^{12} - 64 p^{5} T^{13} + 53 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 + 14 T + 170 T^{2} + 1454 T^{3} + 11888 T^{4} + 78870 T^{5} + 496918 T^{6} + 2681430 T^{7} + 13842846 T^{8} + 2681430 p T^{9} + 496918 p^{2} T^{10} + 78870 p^{3} T^{11} + 11888 p^{4} T^{12} + 1454 p^{5} T^{13} + 170 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 8 T + 69 T^{2} - 640 T^{3} + 5258 T^{4} - 32016 T^{5} + 234283 T^{6} - 1390520 T^{7} + 7909578 T^{8} - 1390520 p T^{9} + 234283 p^{2} T^{10} - 32016 p^{3} T^{11} + 5258 p^{4} T^{12} - 640 p^{5} T^{13} + 69 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 4 T - 12 T^{2} + 172 T^{3} + 2372 T^{4} - 1692 T^{5} - 45204 T^{6} - 18636 T^{7} + 6125910 T^{8} - 18636 p T^{9} - 45204 p^{2} T^{10} - 1692 p^{3} T^{11} + 2372 p^{4} T^{12} + 172 p^{5} T^{13} - 12 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 2 T + 191 T^{2} - 282 T^{3} + 16837 T^{4} - 9752 T^{5} + 959494 T^{6} + 192572 T^{7} + 42746654 T^{8} + 192572 p T^{9} + 959494 p^{2} T^{10} - 9752 p^{3} T^{11} + 16837 p^{4} T^{12} - 282 p^{5} T^{13} + 191 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 2 T + 149 T^{2} + 72 T^{3} + 10982 T^{4} + 34540 T^{5} + 531475 T^{6} + 3068794 T^{7} + 22437618 T^{8} + 3068794 p T^{9} + 531475 p^{2} T^{10} + 34540 p^{3} T^{11} + 10982 p^{4} T^{12} + 72 p^{5} T^{13} + 149 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 4 p T^{2} + 1860 T^{3} + 20683 T^{4} + 166890 T^{5} + 1504675 T^{6} + 10715322 T^{7} + 1738954 p T^{8} + 10715322 p T^{9} + 1504675 p^{2} T^{10} + 166890 p^{3} T^{11} + 20683 p^{4} T^{12} + 1860 p^{5} T^{13} + 4 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 4 T + 214 T^{2} + 704 T^{3} + 19568 T^{4} + 29592 T^{5} + 1102106 T^{6} - 1013740 T^{7} + 54838974 T^{8} - 1013740 p T^{9} + 1102106 p^{2} T^{10} + 29592 p^{3} T^{11} + 19568 p^{4} T^{12} + 704 p^{5} T^{13} + 214 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 18 T + 326 T^{2} - 4506 T^{3} + 51520 T^{4} - 557562 T^{5} + 5314762 T^{6} - 46070178 T^{7} + 378142430 T^{8} - 46070178 p T^{9} + 5314762 p^{2} T^{10} - 557562 p^{3} T^{11} + 51520 p^{4} T^{12} - 4506 p^{5} T^{13} + 326 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 12 T + 277 T^{2} - 3136 T^{3} + 39870 T^{4} - 406200 T^{5} + 3782003 T^{6} - 34750508 T^{7} + 263133730 T^{8} - 34750508 p T^{9} + 3782003 p^{2} T^{10} - 406200 p^{3} T^{11} + 39870 p^{4} T^{12} - 3136 p^{5} T^{13} + 277 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 26 T + 451 T^{2} - 6050 T^{3} + 75913 T^{4} - 854712 T^{5} + 8812957 T^{6} - 80168870 T^{7} + 682181179 T^{8} - 80168870 p T^{9} + 8812957 p^{2} T^{10} - 854712 p^{3} T^{11} + 75913 p^{4} T^{12} - 6050 p^{5} T^{13} + 451 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 24 T + 686 T^{2} - 11044 T^{3} + 184584 T^{4} - 2232828 T^{5} + 27109682 T^{6} - 257392272 T^{7} + 2428813230 T^{8} - 257392272 p T^{9} + 27109682 p^{2} T^{10} - 2232828 p^{3} T^{11} + 184584 p^{4} T^{12} - 11044 p^{5} T^{13} + 686 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 14 T + 300 T^{2} + 3234 T^{3} + 50772 T^{4} + 457014 T^{5} + 5630548 T^{6} + 610346 p T^{7} + 477196566 T^{8} + 610346 p^{2} T^{9} + 5630548 p^{2} T^{10} + 457014 p^{3} T^{11} + 50772 p^{4} T^{12} + 3234 p^{5} T^{13} + 300 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T + 400 T^{2} - 4042 T^{3} + 84364 T^{4} - 769890 T^{5} + 11509232 T^{6} - 91087970 T^{7} + 1083990374 T^{8} - 91087970 p T^{9} + 11509232 p^{2} T^{10} - 769890 p^{3} T^{11} + 84364 p^{4} T^{12} - 4042 p^{5} T^{13} + 400 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 40 T + 916 T^{2} + 15056 T^{3} + 213012 T^{4} + 2711520 T^{5} + 31421068 T^{6} + 325262680 T^{7} + 3089294838 T^{8} + 325262680 p T^{9} + 31421068 p^{2} T^{10} + 2711520 p^{3} T^{11} + 213012 p^{4} T^{12} + 15056 p^{5} T^{13} + 916 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 34 T + 794 T^{2} - 14230 T^{3} + 218299 T^{4} - 2889828 T^{5} + 34814851 T^{6} - 375825900 T^{7} + 3720460370 T^{8} - 375825900 p T^{9} + 34814851 p^{2} T^{10} - 2889828 p^{3} T^{11} + 218299 p^{4} T^{12} - 14230 p^{5} T^{13} + 794 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 30 T + 893 T^{2} - 16084 T^{3} + 286266 T^{4} - 3832848 T^{5} + 51219155 T^{6} - 551902430 T^{7} + 5992632938 T^{8} - 551902430 p T^{9} + 51219155 p^{2} T^{10} - 3832848 p^{3} T^{11} + 286266 p^{4} T^{12} - 16084 p^{5} T^{13} + 893 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.90458636026905220803120738912, −3.67450135568525205193518487457, −3.60937052124579437279199746164, −3.58670327736621200527163270118, −3.52534545507026369945491233549, −3.36305665104416248127961615640, −3.09276178305079768419663398661, −2.92319217425504110822603056912, −2.65670989075033562603567888800, −2.60968081231717799704858326995, −2.24972894538789046472460918091, −2.22887168477300029850118123728, −2.22533892996369667379036589032, −1.80901281345292221085470937463, −1.79080821953127293385960464746, −1.69678880257147556323904755680, −1.59030735987841178139386653254, −1.47834099910361017075999504871, −1.14884792375178383344801016190, −0.983878420232591295657335447188, −0.807133543343497993372000165724, −0.59146369806927440250452264530, −0.56006473139472308493909671056, −0.47509649146166903336571070911, −0.36888289924507259664087467196, 0.36888289924507259664087467196, 0.47509649146166903336571070911, 0.56006473139472308493909671056, 0.59146369806927440250452264530, 0.807133543343497993372000165724, 0.983878420232591295657335447188, 1.14884792375178383344801016190, 1.47834099910361017075999504871, 1.59030735987841178139386653254, 1.69678880257147556323904755680, 1.79080821953127293385960464746, 1.80901281345292221085470937463, 2.22533892996369667379036589032, 2.22887168477300029850118123728, 2.24972894538789046472460918091, 2.60968081231717799704858326995, 2.65670989075033562603567888800, 2.92319217425504110822603056912, 3.09276178305079768419663398661, 3.36305665104416248127961615640, 3.52534545507026369945491233549, 3.58670327736621200527163270118, 3.60937052124579437279199746164, 3.67450135568525205193518487457, 3.90458636026905220803120738912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.