Dirichlet series
| L(s) = 1 | + 2·3-s + 4-s − 7-s − 5·8-s + 9-s − 3·11-s + 2·12-s − 6·13-s − 16-s + 10·17-s + 6·19-s − 2·21-s + 10·23-s − 10·24-s − 28-s + 3·31-s − 5·32-s − 6·33-s + 36-s + 19·37-s − 12·39-s − 25·41-s + 4·43-s − 3·44-s − 15·47-s − 2·48-s + 18·49-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.377·7-s − 1.76·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.66·13-s − 1/4·16-s + 2.42·17-s + 1.37·19-s − 0.436·21-s + 2.08·23-s − 2.04·24-s − 0.188·28-s + 0.538·31-s − 0.883·32-s − 1.04·33-s + 1/6·36-s + 3.12·37-s − 1.92·39-s − 3.90·41-s + 0.609·43-s − 0.452·44-s − 2.18·47-s − 0.288·48-s + 18/7·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.54689\times 10^{6}\) |
| Root analytic conductor: | \(2.56664\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06991074978\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06991074978\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | \( 1 \) | |
| 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
−4.39735155938394862145143882000, −4.24219274713163718573979944734, −4.03216789647915869915714281821, −3.79728187860782538046485939383, −3.79301327269138391813260019062, −3.58400679863435834684581048486, −3.50720843450651045119153615217, −3.43813190745573116827641858160, −3.39263954382239583992676463162, −3.25141994012966842411687633599, −2.85452338961852613755180157207, −2.78868754991763172346862368031, −2.78628428175166812179378958449, −2.67208207207784548339927029294, −2.33679538605375073096955659830, −2.28502476274412261980154198528, −2.25584792631942894207701098680, −2.10220126901709932379634924783, −2.06301804243813749075509055268, −1.26253915589287115210163160930, −1.18616893066572821877535260385, −1.09723895388584241691885316113, −0.880362008215214541359063878965, −0.793831081968000064909716372896, −0.02719591954086148918470482113, 0.02719591954086148918470482113, 0.793831081968000064909716372896, 0.880362008215214541359063878965, 1.09723895388584241691885316113, 1.18616893066572821877535260385, 1.26253915589287115210163160930, 2.06301804243813749075509055268, 2.10220126901709932379634924783, 2.25584792631942894207701098680, 2.28502476274412261980154198528, 2.33679538605375073096955659830, 2.67208207207784548339927029294, 2.78628428175166812179378958449, 2.78868754991763172346862368031, 2.85452338961852613755180157207, 3.25141994012966842411687633599, 3.39263954382239583992676463162, 3.43813190745573116827641858160, 3.50720843450651045119153615217, 3.58400679863435834684581048486, 3.79301327269138391813260019062, 3.79728187860782538046485939383, 4.03216789647915869915714281821, 4.24219274713163718573979944734, 4.39735155938394862145143882000
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.