Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s + 16·7-s + 7·9-s − 4·11-s − 2·12-s − 2·13-s + 32·14-s − 4·17-s + 14·18-s − 10·19-s − 32·21-s − 8·22-s − 12·23-s − 4·26-s − 6·27-s + 16·28-s − 10·29-s + 6·31-s − 2·32-s + 8·33-s − 8·34-s + 7·36-s − 4·37-s − 20·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s + 6.04·7-s + 7/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 8.55·14-s − 0.970·17-s + 3.29·18-s − 2.29·19-s − 6.98·21-s − 1.70·22-s − 2.50·23-s − 0.784·26-s − 1.15·27-s + 3.02·28-s − 1.85·29-s + 1.07·31-s − 0.353·32-s + 1.39·33-s − 1.37·34-s + 7/6·36-s − 0.657·37-s − 3.24·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(252.195\) |
| Root analytic conductor: | \(1.41289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.831694951\) |
| \(L(\frac12)\) | \(\approx\) | \(3.831694951\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | \( 1 \) | |
| good | 3 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 - 8 T + 6 p T^{2} - 160 T^{3} + 471 T^{4} - 160 p T^{5} + 6 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( 1 + 4 T - 25 T^{2} - 80 T^{3} + 260 T^{4} + 752 T^{5} + 123 p T^{6} - 210 p T^{7} - 44765 T^{8} - 210 p^{2} T^{9} + 123 p^{3} T^{10} + 752 p^{3} T^{11} + 260 p^{4} T^{12} - 80 p^{5} T^{13} - 25 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T + 7 T^{2} - 34 T^{3} + 16 T^{4} - 68 T^{5} + 985 T^{6} - 2342 T^{7} - 9421 T^{8} - 2342 p T^{9} + 985 p^{2} T^{10} - 68 p^{3} T^{11} + 16 p^{4} T^{12} - 34 p^{5} T^{13} + 7 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T - p T^{2} - 132 T^{3} + 111 T^{4} + 364 T^{5} - 6555 T^{6} + 8564 T^{7} + 249544 T^{8} + 8564 p T^{9} - 6555 p^{2} T^{10} + 364 p^{3} T^{11} + 111 p^{4} T^{12} - 132 p^{5} T^{13} - p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 10 T + 22 T^{2} + 30 T^{3} + 1023 T^{4} + 4210 T^{5} - 4736 T^{6} + 17200 T^{7} + 424105 T^{8} + 17200 p T^{9} - 4736 p^{2} T^{10} + 4210 p^{3} T^{11} + 1023 p^{4} T^{12} + 30 p^{5} T^{13} + 22 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 12 T + 17 T^{2} - 244 T^{3} - 304 T^{4} + 6472 T^{5} + 29135 T^{6} + 23748 T^{7} - 173281 T^{8} + 23748 p T^{9} + 29135 p^{2} T^{10} + 6472 p^{3} T^{11} - 304 p^{4} T^{12} - 244 p^{5} T^{13} + 17 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 10 T + 47 T^{2} + 470 T^{3} + 4888 T^{4} + 28460 T^{5} + 139689 T^{6} + 999650 T^{7} + 6593475 T^{8} + 999650 p T^{9} + 139689 p^{2} T^{10} + 28460 p^{3} T^{11} + 4888 p^{4} T^{12} + 470 p^{5} T^{13} + 47 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 113520 p T^{9} - 9417 p^{2} T^{10} - 4048 p^{3} T^{11} + 1840 p^{4} T^{12} - 110 p^{5} T^{13} + 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 4 T - 7 T^{2} - 6 p T^{3} + 371 T^{4} - 7606 T^{5} - 27105 T^{6} - 1656 T^{7} + 3727704 T^{8} - 1656 p T^{9} - 27105 p^{2} T^{10} - 7606 p^{3} T^{11} + 371 p^{4} T^{12} - 6 p^{6} T^{13} - 7 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 14 T + 45 T^{2} + 130 T^{3} + 2680 T^{4} + 7652 T^{5} - 46977 T^{6} - 402900 T^{7} - 2944785 T^{8} - 402900 p T^{9} - 46977 p^{2} T^{10} + 7652 p^{3} T^{11} + 2680 p^{4} T^{12} + 130 p^{5} T^{13} + 45 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 14 T + 158 T^{2} - 1080 T^{3} + 7971 T^{4} - 1080 p T^{5} + 158 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 24 T + 313 T^{2} + 2808 T^{3} + 18636 T^{4} + 1392 p T^{5} - 279085 T^{6} - 6389496 T^{7} - 55226161 T^{8} - 6389496 p T^{9} - 279085 p^{2} T^{10} + 1392 p^{4} T^{11} + 18636 p^{4} T^{12} + 2808 p^{5} T^{13} + 313 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T - 58 T^{2} + 116 T^{3} + 127 p T^{4} - 21448 T^{5} - 130860 T^{6} - 424352 T^{7} + 13016709 T^{8} - 424352 p T^{9} - 130860 p^{2} T^{10} - 21448 p^{3} T^{11} + 127 p^{5} T^{12} + 116 p^{5} T^{13} - 58 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 p T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 324004 p^{2} T^{10} + 3563 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 1508020 p T^{9} - 186192 p^{2} T^{10} + 25262 p^{3} T^{11} + 5215 p^{4} T^{12} - 30 p^{5} T^{13} + 30 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 4 T - 137 T^{2} - 192 T^{3} + 10676 T^{4} + 15424 T^{5} - 242695 T^{6} + 235354 T^{7} + 7588819 T^{8} + 235354 p T^{9} - 242695 p^{2} T^{10} + 15424 p^{3} T^{11} + 10676 p^{4} T^{12} - 192 p^{5} T^{13} - 137 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 34 T + 425 T^{2} + 2570 T^{3} + 16960 T^{4} + 219512 T^{5} + 2212323 T^{6} + 19311440 T^{7} + 171308535 T^{8} + 19311440 p T^{9} + 2212323 p^{2} T^{10} + 219512 p^{3} T^{11} + 16960 p^{4} T^{12} + 2570 p^{5} T^{13} + 425 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 12 T - 38 T^{2} - 1764 T^{3} - 9549 T^{4} + 121692 T^{5} + 1406420 T^{6} - 2622672 T^{7} - 104550091 T^{8} - 2622672 p T^{9} + 1406420 p^{2} T^{10} + 121692 p^{3} T^{11} - 9549 p^{4} T^{12} - 1764 p^{5} T^{13} - 38 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 138 T^{2} - 1500 T^{3} + 4403 T^{4} + 205200 T^{5} + 1430844 T^{6} - 7779600 T^{7} - 206126795 T^{8} - 7779600 p T^{9} + 1430844 p^{2} T^{10} + 205200 p^{3} T^{11} + 4403 p^{4} T^{12} - 1500 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 + 32 T + 427 T^{2} + 2916 T^{3} + 11596 T^{4} + 29072 T^{5} - 1136915 T^{6} - 33374062 T^{7} - 418943541 T^{8} - 33374062 p T^{9} - 1136915 p^{2} T^{10} + 29072 p^{3} T^{11} + 11596 p^{4} T^{12} + 2916 p^{5} T^{13} + 427 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 1290 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 4 T - 62 T^{2} - 1692 T^{3} + 3191 T^{4} - 2396 T^{5} + 172940 T^{6} - 2472896 T^{7} + 139003669 T^{8} - 2472896 p T^{9} + 172940 p^{2} T^{10} - 2396 p^{3} T^{11} + 3191 p^{4} T^{12} - 1692 p^{5} T^{13} - 62 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.49093703479943886321986074602, −5.43372771642662983347066278302, −4.98571559554719292923555122886, −4.90462421876633223221319847536, −4.68597308322830525227790415343, −4.59263634450102664309031021697, −4.52686115532177129638187565949, −4.49992812249628717753891832154, −4.36400332421360385316671949810, −4.33737840807449090638038776995, −4.02254084289118645323962448667, −4.00880648930274704342075423455, −4.00010403702497186433442521473, −3.36840450275829870156031297583, −3.13440360938131157882986120569, −2.80198363581217135867929795994, −2.73217128410761034916478451304, −2.64094438855262464013720547792, −2.00150833797728497691990039748, −1.83534057796275142764082466626, −1.66134306922778889926478198438, −1.66041690072960894843552685042, −1.64428550327775767555676573910, −1.56058883963570202201840204955, −0.41756539097360934869493111520, 0.41756539097360934869493111520, 1.56058883963570202201840204955, 1.64428550327775767555676573910, 1.66041690072960894843552685042, 1.66134306922778889926478198438, 1.83534057796275142764082466626, 2.00150833797728497691990039748, 2.64094438855262464013720547792, 2.73217128410761034916478451304, 2.80198363581217135867929795994, 3.13440360938131157882986120569, 3.36840450275829870156031297583, 4.00010403702497186433442521473, 4.00880648930274704342075423455, 4.02254084289118645323962448667, 4.33737840807449090638038776995, 4.36400332421360385316671949810, 4.49992812249628717753891832154, 4.52686115532177129638187565949, 4.59263634450102664309031021697, 4.68597308322830525227790415343, 4.90462421876633223221319847536, 4.98571559554719292923555122886, 5.43372771642662983347066278302, 5.49093703479943886321986074602
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.