Dirichlet series
| L(s) = 1 | − 8·2-s + 32·4-s − 80·8-s − 8·11-s − 8·13-s + 120·16-s + 32·17-s + 64·22-s + 40·23-s + 64·26-s + 144·31-s − 32·32-s − 256·34-s − 160·37-s − 320·41-s + 32·43-s − 256·44-s − 320·46-s + 144·47-s − 256·52-s + 200·53-s + 288·61-s − 1.15e3·62-s − 384·64-s − 80·67-s + 1.02e3·68-s − 280·71-s + ⋯ |
| L(s) = 1 | − 4·2-s + 8·4-s − 10·8-s − 0.727·11-s − 0.615·13-s + 15/2·16-s + 1.88·17-s + 2.90·22-s + 1.73·23-s + 2.46·26-s + 4.64·31-s − 32-s − 7.52·34-s − 4.32·37-s − 7.80·41-s + 0.744·43-s − 5.81·44-s − 6.95·46-s + 3.06·47-s − 4.92·52-s + 3.77·53-s + 4.72·61-s − 18.5·62-s − 6·64-s − 1.19·67-s + 15.0·68-s − 3.94·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48946\times 10^{11}\) |
| Root analytic conductor: | \(5.34887\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.1500824535\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1500824535\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| good | 11 | \( ( 1 + 4 T + 120 T^{2} - 964 T^{3} + 9758 T^{4} - 964 p^{2} T^{5} + 120 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 8 T + 32 T^{2} + 1000 T^{3} + 57188 T^{4} + 379192 T^{5} + 10080 p^{2} T^{6} + 4420344 p T^{7} + 1905455878 T^{8} + 4420344 p^{3} T^{9} + 10080 p^{6} T^{10} + 379192 p^{6} T^{11} + 57188 p^{8} T^{12} + 1000 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 32 T + 512 T^{2} - 13280 T^{3} + 494308 T^{4} - 9125408 T^{5} + 127106560 T^{6} - 2946229728 T^{7} + 67506528198 T^{8} - 2946229728 p^{2} T^{9} + 127106560 p^{4} T^{10} - 9125408 p^{6} T^{11} + 494308 p^{8} T^{12} - 13280 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 1632 T^{2} + 1354756 T^{4} - 767600928 T^{6} + 320946269766 T^{8} - 767600928 p^{4} T^{10} + 1354756 p^{8} T^{12} - 1632 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 - 40 T + 800 T^{2} - 23320 T^{3} + 928156 T^{4} - 22803400 T^{5} + 441522400 T^{6} - 12864455160 T^{7} + 373563445830 T^{8} - 12864455160 p^{2} T^{9} + 441522400 p^{4} T^{10} - 22803400 p^{6} T^{11} + 928156 p^{8} T^{12} - 23320 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 5816 T^{2} + 15494364 T^{4} - 24578727304 T^{6} + 25344087778310 T^{8} - 24578727304 p^{4} T^{10} + 15494364 p^{8} T^{12} - 5816 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 - 72 T + 4816 T^{2} - 187848 T^{3} + 7076706 T^{4} - 187848 p^{2} T^{5} + 4816 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 + 160 T + 12800 T^{2} + 692576 T^{3} + 28393916 T^{4} + 973559072 T^{5} + 32158084608 T^{6} + 1118111280864 T^{7} + 40611983255110 T^{8} + 1118111280864 p^{2} T^{9} + 32158084608 p^{4} T^{10} + 973559072 p^{6} T^{11} + 28393916 p^{8} T^{12} + 692576 p^{10} T^{13} + 12800 p^{12} T^{14} + 160 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 + 160 T + 11944 T^{2} + 591520 T^{3} + 24800530 T^{4} + 591520 p^{2} T^{5} + 11944 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 - 32 T + 512 T^{2} + 44896 T^{3} - 2147132 T^{4} - 116117984 T^{5} + 5822932480 T^{6} - 160684182624 T^{7} + 3183857489478 T^{8} - 160684182624 p^{2} T^{9} + 5822932480 p^{4} T^{10} - 116117984 p^{6} T^{11} - 2147132 p^{8} T^{12} + 44896 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 144 T + 10368 T^{2} - 613776 T^{3} + 40380676 T^{4} - 2550145200 T^{5} + 136914549120 T^{6} - 6692132733360 T^{7} + 317421876290310 T^{8} - 6692132733360 p^{2} T^{9} + 136914549120 p^{4} T^{10} - 2550145200 p^{6} T^{11} + 40380676 p^{8} T^{12} - 613776 p^{10} T^{13} + 10368 p^{12} T^{14} - 144 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 200 T + 20000 T^{2} - 1719464 T^{3} + 146284516 T^{4} - 10449435128 T^{5} + 642474929248 T^{6} - 39041749119576 T^{7} + 2230315686063750 T^{8} - 39041749119576 p^{2} T^{9} + 642474929248 p^{4} T^{10} - 10449435128 p^{6} T^{11} + 146284516 p^{8} T^{12} - 1719464 p^{10} T^{13} + 20000 p^{12} T^{14} - 200 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 616 T^{2} + 1419676 T^{4} + 5207600552 T^{6} - 204890323221242 T^{8} + 5207600552 p^{4} T^{10} + 1419676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 144 T + 14124 T^{2} - 1141872 T^{3} + 82373126 T^{4} - 1141872 p^{2} T^{5} + 14124 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 + 80 T + 3200 T^{2} + 105296 T^{3} - 42100604 T^{4} - 2558251408 T^{5} - 64394556032 T^{6} + 2106545111664 T^{7} + 886628492165190 T^{8} + 2106545111664 p^{2} T^{9} - 64394556032 p^{4} T^{10} - 2558251408 p^{6} T^{11} - 42100604 p^{8} T^{12} + 105296 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 140 T + 26424 T^{2} + 2214100 T^{3} + 216062270 T^{4} + 2214100 p^{2} T^{5} + 26424 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 312 T + 48672 T^{2} + 5493144 T^{3} + 450793636 T^{4} + 23206886280 T^{5} + 386836170336 T^{6} - 68692339480344 T^{7} - 8375186112407610 T^{8} - 68692339480344 p^{2} T^{9} + 386836170336 p^{4} T^{10} + 23206886280 p^{6} T^{11} + 450793636 p^{8} T^{12} + 5493144 p^{10} T^{13} + 48672 p^{12} T^{14} + 312 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 2856 T^{2} - 59627684 T^{4} + 132589015272 T^{6} + 2412904555054278 T^{8} + 132589015272 p^{4} T^{10} - 59627684 p^{8} T^{12} - 2856 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 - 320 T + 51200 T^{2} - 5915840 T^{3} + 553217092 T^{4} - 45528000320 T^{5} + 3742826444800 T^{6} - 317655699460800 T^{7} + 26711748349549254 T^{8} - 317655699460800 p^{2} T^{9} + 3742826444800 p^{4} T^{10} - 45528000320 p^{6} T^{11} + 553217092 p^{8} T^{12} - 5915840 p^{10} T^{13} + 51200 p^{12} T^{14} - 320 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 26800 T^{2} + 323387236 T^{4} - 2700009379600 T^{6} + 20962417894146886 T^{8} - 2700009379600 p^{4} T^{10} + 323387236 p^{8} T^{12} - 26800 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 - 24 T + 288 T^{2} - 197496 T^{3} - 223582556 T^{4} + 4223145048 T^{5} - 17461370016 T^{6} - 856985885448 T^{7} + 24034841148321606 T^{8} - 856985885448 p^{2} T^{9} - 17461370016 p^{4} T^{10} + 4223145048 p^{6} T^{11} - 223582556 p^{8} T^{12} - 197496 p^{10} T^{13} + 288 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−3.97517641505770372275475293634, −3.89674526820651553616479483299, −3.62576497981924301619271650563, −3.50414806866800495101075450709, −3.42547667127964384727123797252, −3.19297809913605966529496164850, −3.18322731565955831793100404842, −2.93529251967143498411733825290, −2.78963337018509636923494730274, −2.72628163168141357247816899852, −2.57828989988562907291425839411, −2.33732703858987073423402777448, −2.28755961755530212796706161603, −2.13892191617676596249654153529, −1.76165380423896199032233714976, −1.73787168884313443880635101389, −1.48835637141135122108096506599, −1.44799408187290145096079902108, −1.29034883580571915319546056949, −1.02470245784063802596988546587, −0.916937670212761001464004916935, −0.76497109583220262679825893210, −0.49478843386538419788328391411, −0.30264618496058805028190434627, −0.10887724806364564108526052772, 0.10887724806364564108526052772, 0.30264618496058805028190434627, 0.49478843386538419788328391411, 0.76497109583220262679825893210, 0.916937670212761001464004916935, 1.02470245784063802596988546587, 1.29034883580571915319546056949, 1.44799408187290145096079902108, 1.48835637141135122108096506599, 1.73787168884313443880635101389, 1.76165380423896199032233714976, 2.13892191617676596249654153529, 2.28755961755530212796706161603, 2.33732703858987073423402777448, 2.57828989988562907291425839411, 2.72628163168141357247816899852, 2.78963337018509636923494730274, 2.93529251967143498411733825290, 3.18322731565955831793100404842, 3.19297809913605966529496164850, 3.42547667127964384727123797252, 3.50414806866800495101075450709, 3.62576497981924301619271650563, 3.89674526820651553616479483299, 3.97517641505770372275475293634
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.