Dirichlet series
| L(s) = 1 | + 4-s − 2·5-s + 7-s − 5·8-s + 3·11-s + 6·13-s − 16-s + 10·17-s + 6·19-s − 2·20-s + 10·23-s + 25-s + 28-s + 3·31-s − 5·32-s − 2·35-s − 19·37-s + 10·40-s + 25·41-s − 4·43-s + 3·44-s − 15·47-s + 18·49-s + 6·52-s − 7·53-s − 6·55-s − 5·56-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.76·8-s + 0.904·11-s + 1.66·13-s − 1/4·16-s + 2.42·17-s + 1.37·19-s − 0.447·20-s + 2.08·23-s + 1/5·25-s + 0.188·28-s + 0.538·31-s − 0.883·32-s − 0.338·35-s − 3.12·37-s + 1.58·40-s + 3.90·41-s − 0.609·43-s + 0.452·44-s − 2.18·47-s + 18/7·49-s + 0.832·52-s − 0.961·53-s − 0.809·55-s − 0.668·56-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(59574.2\) |
| Root analytic conductor: | \(1.98811\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.096278354\) |
| \(L(\frac12)\) | \(\approx\) | \(2.096278354\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 3 T + 8 T^{2} - T^{3} - 85 T^{4} - p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( ( 1 - 3 T^{2} + 9 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )( 1 + p T^{2} + 5 T^{3} - T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{8} ) \) |
| 7 | \( 1 - T - 17 T^{2} + 4 p T^{3} + 61 T^{4} - 356 T^{5} + 670 T^{6} + 1439 T^{7} - 8371 T^{8} + 1439 p T^{9} + 670 p^{2} T^{10} - 356 p^{3} T^{11} + 61 p^{4} T^{12} + 4 p^{6} T^{13} - 17 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 6 T + p T^{2} - 48 T^{3} + 458 T^{4} - 66 p T^{5} - 3109 T^{6} + 5886 T^{7} + 35023 T^{8} + 5886 p T^{9} - 3109 p^{2} T^{10} - 66 p^{4} T^{11} + 458 p^{4} T^{12} - 48 p^{5} T^{13} + p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 - 5 T + 8 T^{2} + 45 T^{3} - 361 T^{4} + 45 p T^{5} + 8 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( 1 - 6 T - 29 T^{2} + 180 T^{3} + 12 p T^{4} - 1410 T^{5} + 5459 T^{6} - 8316 T^{7} - 135877 T^{8} - 8316 p T^{9} + 5459 p^{2} T^{10} - 1410 p^{3} T^{11} + 12 p^{5} T^{12} + 180 p^{5} T^{13} - 29 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 5 T + 91 T^{2} - 340 T^{3} + 3127 T^{4} - 340 p T^{5} + 91 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 41 T^{2} + 50 T^{3} + 90 T^{4} - 7910 T^{5} + 39721 T^{6} + 144550 T^{7} - 1367641 T^{8} + 144550 p T^{9} + 39721 p^{2} T^{10} - 7910 p^{3} T^{11} + 90 p^{4} T^{12} + 50 p^{5} T^{13} - 41 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 - 3 T - 51 T^{2} + 125 T^{3} + 1383 T^{4} - 3250 T^{5} - 9974 T^{6} + 2088 p T^{7} - 789407 T^{8} + 2088 p^{2} T^{9} - 9974 p^{2} T^{10} - 3250 p^{3} T^{11} + 1383 p^{4} T^{12} + 125 p^{5} T^{13} - 51 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 19 T + 181 T^{2} + 1667 T^{3} + 16711 T^{4} + 129422 T^{5} + 823180 T^{6} + 5794390 T^{7} + 39506327 T^{8} + 5794390 p T^{9} + 823180 p^{2} T^{10} + 129422 p^{3} T^{11} + 16711 p^{4} T^{12} + 1667 p^{5} T^{13} + 181 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 25 T + 245 T^{2} - 1010 T^{3} - 1171 T^{4} + 27770 T^{5} + 9500 T^{6} - 1860265 T^{7} + 16785431 T^{8} - 1860265 p T^{9} + 9500 p^{2} T^{10} + 27770 p^{3} T^{11} - 1171 p^{4} T^{12} - 1010 p^{5} T^{13} + 245 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 2 T + 80 T^{2} + 195 T^{3} + 5043 T^{4} + 195 p T^{5} + 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 15 T + 29 T^{2} - 980 T^{3} - 7243 T^{4} + 610 p T^{5} + 474542 T^{6} + 134235 T^{7} - 16187995 T^{8} + 134235 p T^{9} + 474542 p^{2} T^{10} + 610 p^{4} T^{11} - 7243 p^{4} T^{12} - 980 p^{5} T^{13} + 29 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 7 T - 120 T^{2} - 432 T^{3} + 5474 T^{4} + 3321 T^{5} + 274274 T^{6} + 446086 T^{7} - 33313149 T^{8} + 446086 p T^{9} + 274274 p^{2} T^{10} + 3321 p^{3} T^{11} + 5474 p^{4} T^{12} - 432 p^{5} T^{13} - 120 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 35 T + 515 T^{2} + 4925 T^{3} + 45479 T^{4} + 408430 T^{5} + 3276960 T^{6} + 29360880 T^{7} + 255959881 T^{8} + 29360880 p T^{9} + 3276960 p^{2} T^{10} + 408430 p^{3} T^{11} + 45479 p^{4} T^{12} + 4925 p^{5} T^{13} + 515 p^{6} T^{14} + 35 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 21 T + 120 T^{2} + 665 T^{3} - 8580 T^{4} - 32368 T^{5} + 553498 T^{6} + 70560 p T^{7} - 1397405 p T^{8} + 70560 p^{2} T^{9} + 553498 p^{2} T^{10} - 32368 p^{3} T^{11} - 8580 p^{4} T^{12} + 665 p^{5} T^{13} + 120 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 13 T + 132 T^{2} + 845 T^{3} + 5331 T^{4} + 845 p T^{5} + 132 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 25 T + 205 T^{2} + 75 T^{3} - 11801 T^{4} - 111650 T^{5} - 105920 T^{6} + 10613650 T^{7} + 139153461 T^{8} + 10613650 p T^{9} - 105920 p^{2} T^{10} - 111650 p^{3} T^{11} - 11801 p^{4} T^{12} + 75 p^{5} T^{13} + 205 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - T - 116 T^{2} - 1465 T^{3} + 5146 T^{4} + 194212 T^{5} + 1001200 T^{6} - 6204182 T^{7} - 161318173 T^{8} - 6204182 p T^{9} + 1001200 p^{2} T^{10} + 194212 p^{3} T^{11} + 5146 p^{4} T^{12} - 1465 p^{5} T^{13} - 116 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 30 T + 329 T^{2} - 2610 T^{3} + 38585 T^{4} - 496770 T^{5} + 4377211 T^{6} - 41029230 T^{7} + 403939204 T^{8} - 41029230 p T^{9} + 4377211 p^{2} T^{10} - 496770 p^{3} T^{11} + 38585 p^{4} T^{12} - 2610 p^{5} T^{13} + 329 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 11 T + 193 T^{2} + 2123 T^{3} + 22473 T^{4} + 108768 T^{5} + 1216536 T^{6} + 2475814 T^{7} + 23169933 T^{8} + 2475814 p T^{9} + 1216536 p^{2} T^{10} + 108768 p^{3} T^{11} + 22473 p^{4} T^{12} + 2123 p^{5} T^{13} + 193 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 16 T + 401 T^{2} + 4140 T^{3} + 55265 T^{4} + 4140 p T^{5} + 401 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 5 T - 214 T^{2} + 710 T^{3} + 19262 T^{4} - 67395 T^{5} + 489658 T^{6} + 1169400 T^{7} - 164725745 T^{8} + 1169400 p T^{9} + 489658 p^{2} T^{10} - 67395 p^{3} T^{11} + 19262 p^{4} T^{12} + 710 p^{5} T^{13} - 214 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.70698846862941386410658304183, −4.65682539411112667291245501807, −4.57293984215362689039624955035, −4.55040782769677690848656899267, −4.34200598694307553073981287368, −3.90945465482265295478297833432, −3.78721814042650933438278099571, −3.65637003258953638332978896279, −3.62442213242878274541951601250, −3.59519868642309678566868570090, −3.37800946996720619674200903545, −3.18727888059829190936647715776, −2.93201043973810625499773282450, −2.90303410377033029125188035489, −2.85124327743109699521355329062, −2.76568968074508321943920098542, −2.43168899833391542021053868103, −2.02118384854476781012478782730, −1.95867362279268281919616499328, −1.58828431791940503924612498044, −1.30525306846147425815365892127, −1.20972652051272957310799304131, −1.20002819559178525208723077006, −0.841960646544957807909112908490, −0.24695823321067149268394806236, 0.24695823321067149268394806236, 0.841960646544957807909112908490, 1.20002819559178525208723077006, 1.20972652051272957310799304131, 1.30525306846147425815365892127, 1.58828431791940503924612498044, 1.95867362279268281919616499328, 2.02118384854476781012478782730, 2.43168899833391542021053868103, 2.76568968074508321943920098542, 2.85124327743109699521355329062, 2.90303410377033029125188035489, 2.93201043973810625499773282450, 3.18727888059829190936647715776, 3.37800946996720619674200903545, 3.59519868642309678566868570090, 3.62442213242878274541951601250, 3.65637003258953638332978896279, 3.78721814042650933438278099571, 3.90945465482265295478297833432, 4.34200598694307553073981287368, 4.55040782769677690848656899267, 4.57293984215362689039624955035, 4.65682539411112667291245501807, 4.70698846862941386410658304183
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.