Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s + 4·11-s − 8·17-s − 8·22-s − 2·25-s + 10·29-s − 14·31-s + 2·32-s + 16·34-s − 8·41-s + 4·44-s + 20·47-s − 4·49-s + 4·50-s + 30·53-s − 20·58-s − 20·59-s + 20·61-s + 28·62-s − 4·64-s − 56·67-s − 8·68-s + 20·71-s − 10·73-s − 20·79-s + 16·82-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.20·11-s − 1.94·17-s − 1.70·22-s − 2/5·25-s + 1.85·29-s − 2.51·31-s + 0.353·32-s + 2.74·34-s − 1.24·41-s + 0.603·44-s + 2.91·47-s − 4/7·49-s + 0.565·50-s + 4.12·53-s − 2.62·58-s − 2.60·59-s + 2.56·61-s + 3.55·62-s − 1/2·64-s − 6.84·67-s − 0.970·68-s + 2.37·71-s − 1.17·73-s − 2.25·79-s + 1.76·82-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(39.0425\) |
| Root analytic conductor: | \(1.25739\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1989938316\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1989938316\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| 11 | \( 1 - 4 T + 4 p T^{3} - 226 T^{4} + 4 p^{2} T^{5} - 4 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 + 2 T^{2} + 4 p T^{3} - T^{4} + 8 p T^{5} + 148 T^{6} + 16 p T^{7} + 341 T^{8} + 16 p^{2} T^{9} + 148 p^{2} T^{10} + 8 p^{4} T^{11} - p^{4} T^{12} + 4 p^{6} T^{13} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 4 T^{2} + 40 T^{3} + 57 T^{4} + 160 T^{5} + 1052 T^{6} + 2020 T^{7} + 4565 T^{8} + 2020 p T^{9} + 1052 p^{2} T^{10} + 160 p^{3} T^{11} + 57 p^{4} T^{12} + 40 p^{5} T^{13} + 4 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 18 T^{2} - 140 T^{3} + 15 T^{4} - 2520 T^{5} + 6398 T^{6} - 7840 T^{7} + 157849 T^{8} - 7840 p T^{9} + 6398 p^{2} T^{10} - 2520 p^{3} T^{11} + 15 p^{4} T^{12} - 140 p^{5} T^{13} + 18 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 8 T + 14 T^{2} + 4 p T^{3} + 863 T^{4} + 3448 T^{5} + 678 p T^{6} + 56904 T^{7} + 246073 T^{8} + 56904 p T^{9} + 678 p^{3} T^{10} + 3448 p^{3} T^{11} + 863 p^{4} T^{12} + 4 p^{6} T^{13} + 14 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 30 T^{2} - 260 T^{3} + 279 T^{4} - 7800 T^{5} + 27830 T^{6} - 80080 T^{7} + 888361 T^{8} - 80080 p T^{9} + 27830 p^{2} T^{10} - 7800 p^{3} T^{11} + 279 p^{4} T^{12} - 260 p^{5} T^{13} + 30 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 - 96 T^{2} + 4612 T^{4} - 151328 T^{6} + 3857990 T^{8} - 151328 p^{2} T^{10} + 4612 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 - 10 T + 57 T^{2} - 460 T^{3} + 4203 T^{4} - 24760 T^{5} + 119139 T^{6} - 824650 T^{7} + 5236800 T^{8} - 824650 p T^{9} + 119139 p^{2} T^{10} - 24760 p^{3} T^{11} + 4203 p^{4} T^{12} - 460 p^{5} T^{13} + 57 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 14 T + 85 T^{2} - 20 T^{3} - 125 p T^{4} - 29468 T^{5} - 66917 T^{6} + 526790 T^{7} + 5550740 T^{8} + 526790 p T^{9} - 66917 p^{2} T^{10} - 29468 p^{3} T^{11} - 125 p^{5} T^{12} - 20 p^{5} T^{13} + 85 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 6 T^{2} + 2847 T^{4} - 52622 T^{6} + 3446025 T^{8} - 52622 p^{2} T^{10} + 2847 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 + 8 T - 84 T^{2} - 588 T^{3} + 2751 T^{4} + 16704 T^{5} + 47004 T^{6} - 114664 T^{7} - 5467983 T^{8} - 114664 p T^{9} + 47004 p^{2} T^{10} + 16704 p^{3} T^{11} + 2751 p^{4} T^{12} - 588 p^{5} T^{13} - 84 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 156 T^{2} + 14432 T^{4} - 933508 T^{6} + 46038510 T^{8} - 933508 p^{2} T^{10} + 14432 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 20 T + 236 T^{2} - 1340 T^{3} + 3167 T^{4} + 22340 T^{5} + 28938 T^{6} - 2709720 T^{7} + 32145805 T^{8} - 2709720 p T^{9} + 28938 p^{2} T^{10} + 22340 p^{3} T^{11} + 3167 p^{4} T^{12} - 1340 p^{5} T^{13} + 236 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 30 T + 569 T^{2} - 6900 T^{3} + 58587 T^{4} - 263880 T^{5} - 611693 T^{6} + 24924930 T^{7} - 240874240 T^{8} + 24924930 p T^{9} - 611693 p^{2} T^{10} - 263880 p^{3} T^{11} + 58587 p^{4} T^{12} - 6900 p^{5} T^{13} + 569 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 20 T + 336 T^{2} + 3400 T^{3} + 24405 T^{4} + 60580 T^{5} - 1156836 T^{6} - 19551820 T^{7} - 192630411 T^{8} - 19551820 p T^{9} - 1156836 p^{2} T^{10} + 60580 p^{3} T^{11} + 24405 p^{4} T^{12} + 3400 p^{5} T^{13} + 336 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 20 T + 150 T^{2} - 1200 T^{3} + 11779 T^{4} - 45580 T^{5} - 36400 T^{6} + 518400 T^{7} - 3011179 T^{8} + 518400 p T^{9} - 36400 p^{2} T^{10} - 45580 p^{3} T^{11} + 11779 p^{4} T^{12} - 1200 p^{5} T^{13} + 150 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 28 T + 452 T^{2} + 5380 T^{3} + 49986 T^{4} + 5380 p T^{5} + 452 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 20 T + 260 T^{2} - 1440 T^{3} - 401 T^{4} + 121620 T^{5} - 777730 T^{6} + 1388800 T^{7} + 34077661 T^{8} + 1388800 p T^{9} - 777730 p^{2} T^{10} + 121620 p^{3} T^{11} - 401 p^{4} T^{12} - 1440 p^{5} T^{13} + 260 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 10 T - 141 T^{2} - 760 T^{3} + 23247 T^{4} + 55820 T^{5} - 2381703 T^{6} - 2640110 T^{7} + 176215880 T^{8} - 2640110 p T^{9} - 2381703 p^{2} T^{10} + 55820 p^{3} T^{11} + 23247 p^{4} T^{12} - 760 p^{5} T^{13} - 141 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 20 T + 56 T^{2} + 40 T^{3} + 24325 T^{4} + 218020 T^{5} - 3204 p T^{6} + 10753340 T^{7} + 280334669 T^{8} + 10753340 p T^{9} - 3204 p^{3} T^{10} + 218020 p^{3} T^{11} + 24325 p^{4} T^{12} + 40 p^{5} T^{13} + 56 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 12 T - 28 T^{2} - 324 T^{3} + 3801 T^{4} - 5388 T^{5} + 255940 T^{6} + 1400868 T^{7} - 37244131 T^{8} + 1400868 p T^{9} + 255940 p^{2} T^{10} - 5388 p^{3} T^{11} + 3801 p^{4} T^{12} - 324 p^{5} T^{13} - 28 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 176 T^{2} + 21966 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 12 T + 34 T^{2} - 708 T^{3} - 6677 T^{4} - 87428 T^{5} - 231884 T^{6} + 1503776 T^{7} + 61442213 T^{8} + 1503776 p T^{9} - 231884 p^{2} T^{10} - 87428 p^{3} T^{11} - 6677 p^{4} T^{12} - 708 p^{5} T^{13} + 34 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.66261795377814675180865065748, −5.66122847317248441553245400727, −5.47916217318311199733899348776, −5.37314331297360228333895986873, −5.21770502743655972690647312270, −4.91640447436882069772226375689, −4.59429916474780294517146392311, −4.43644573944028933885516935960, −4.34445362891450090655216061699, −4.15350585445516980008246620219, −4.07720250240377990758545818019, −4.06937684373197097816460644742, −3.95317870932231002114581438335, −3.41422456535717678874805259078, −3.22402067585037310822995214732, −3.06421135647933448434080744521, −2.84081324638833080186982928766, −2.74253050430348765358163729788, −2.38858336267096770743129472010, −2.24023806054210922617418734513, −1.85087607151462762206876864301, −1.66107112240000205828567178922, −1.23533685471533350310884926587, −1.19449105594262501003521200278, −0.24459902249155168122071795385, 0.24459902249155168122071795385, 1.19449105594262501003521200278, 1.23533685471533350310884926587, 1.66107112240000205828567178922, 1.85087607151462762206876864301, 2.24023806054210922617418734513, 2.38858336267096770743129472010, 2.74253050430348765358163729788, 2.84081324638833080186982928766, 3.06421135647933448434080744521, 3.22402067585037310822995214732, 3.41422456535717678874805259078, 3.95317870932231002114581438335, 4.06937684373197097816460644742, 4.07720250240377990758545818019, 4.15350585445516980008246620219, 4.34445362891450090655216061699, 4.43644573944028933885516935960, 4.59429916474780294517146392311, 4.91640447436882069772226375689, 5.21770502743655972690647312270, 5.37314331297360228333895986873, 5.47916217318311199733899348776, 5.66122847317248441553245400727, 5.66261795377814675180865065748
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.