Dirichlet series
| L(s) = 1 | + 4·2-s + 6·4-s − 2·5-s + 3·7-s − 8·10-s − 12·11-s − 11·13-s + 12·14-s − 15·16-s − 4·17-s − 12·19-s − 12·20-s − 48·22-s + 10·23-s + 3·25-s − 44·26-s + 18·28-s − 2·29-s + 6·31-s − 24·32-s − 16·34-s − 6·35-s − 28·37-s − 48·38-s − 6·43-s − 72·44-s + 40·46-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s − 0.894·5-s + 1.13·7-s − 2.52·10-s − 3.61·11-s − 3.05·13-s + 3.20·14-s − 3.75·16-s − 0.970·17-s − 2.75·19-s − 2.68·20-s − 10.2·22-s + 2.08·23-s + 3/5·25-s − 8.62·26-s + 3.40·28-s − 0.371·29-s + 1.07·31-s − 4.24·32-s − 2.74·34-s − 1.01·35-s − 4.60·37-s − 7.78·38-s − 0.914·43-s − 10.8·44-s + 5.89·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.56501\times 10^{8}\) |
| Root analytic conductor: | \(3.61655\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06142461016\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06142461016\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 3 T + 8 T^{2} - 33 T^{3} + 123 T^{4} - 33 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 + 11 T + 62 T^{2} + 267 T^{3} + 1031 T^{4} + 267 p T^{5} + 62 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 + 2 T + T^{2} + 6 T^{3} + 12 T^{4} + 12 T^{5} + 159 T^{6} + 16 p^{2} T^{7} + 159 T^{8} + 16 p^{3} T^{9} + 159 p^{2} T^{10} + 12 p^{3} T^{11} + 12 p^{4} T^{12} + 6 p^{5} T^{13} + p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 6 T + 34 T^{2} + 109 T^{3} + 439 T^{4} + 109 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 4 T - 38 T^{2} - 210 T^{3} + 807 T^{4} + 5067 T^{5} - 5781 T^{6} - 41251 T^{7} + 39747 T^{8} - 41251 p T^{9} - 5781 p^{2} T^{10} + 5067 p^{3} T^{11} + 807 p^{4} T^{12} - 210 p^{5} T^{13} - 38 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + 6 T + 81 T^{2} + 326 T^{3} + 2343 T^{4} + 326 p T^{5} + 81 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 10 T + 18 T^{2} + 250 T^{3} - 51 p T^{4} - 5235 T^{5} + 41077 T^{6} + 99925 T^{7} - 1499403 T^{8} + 99925 p T^{9} + 41077 p^{2} T^{10} - 5235 p^{3} T^{11} - 51 p^{5} T^{12} + 250 p^{5} T^{13} + 18 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T - 62 T^{2} - 438 T^{3} + 1707 T^{4} + 18759 T^{5} + 38493 T^{6} - 338675 T^{7} - 2208849 T^{8} - 338675 p T^{9} + 38493 p^{2} T^{10} + 18759 p^{3} T^{11} + 1707 p^{4} T^{12} - 438 p^{5} T^{13} - 62 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T - 40 T^{2} + 498 T^{3} - 335 T^{4} - 12051 T^{5} + 37955 T^{6} + 80889 T^{7} - 826703 T^{8} + 80889 p T^{9} + 37955 p^{2} T^{10} - 12051 p^{3} T^{11} - 335 p^{4} T^{12} + 498 p^{5} T^{13} - 40 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 28 T + 376 T^{2} + 3562 T^{3} + 28731 T^{4} + 200055 T^{5} + 1210029 T^{6} + 7124103 T^{7} + 43260595 T^{8} + 7124103 p T^{9} + 1210029 p^{2} T^{10} + 200055 p^{3} T^{11} + 28731 p^{4} T^{12} + 3562 p^{5} T^{13} + 376 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 72 T^{2} - 198 T^{3} + 1259 T^{4} + 11187 T^{5} - 30735 T^{6} - 195921 T^{7} + 3463719 T^{8} - 195921 p T^{9} - 30735 p^{2} T^{10} + 11187 p^{3} T^{11} + 1259 p^{4} T^{12} - 198 p^{5} T^{13} - 72 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 + 6 T - 98 T^{2} - 18 p T^{3} + 5329 T^{4} + 44937 T^{5} - 143171 T^{6} - 901221 T^{7} + 4636105 T^{8} - 901221 p T^{9} - 143171 p^{2} T^{10} + 44937 p^{3} T^{11} + 5329 p^{4} T^{12} - 18 p^{6} T^{13} - 98 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + T - 131 T^{2} + 78 T^{3} + 9356 T^{4} - 11249 T^{5} - 478776 T^{6} + 277502 T^{7} + 21461767 T^{8} + 277502 p T^{9} - 478776 p^{2} T^{10} - 11249 p^{3} T^{11} + 9356 p^{4} T^{12} + 78 p^{5} T^{13} - 131 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 7 T - 119 T^{2} - 1086 T^{3} + 8036 T^{4} + 78157 T^{5} - 258144 T^{6} - 1916824 T^{7} + 10200157 T^{8} - 1916824 p T^{9} - 258144 p^{2} T^{10} + 78157 p^{3} T^{11} + 8036 p^{4} T^{12} - 1086 p^{5} T^{13} - 119 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 2 T - 89 T^{2} + 798 T^{3} + 4938 T^{4} - 63804 T^{5} + 375171 T^{6} + 3093430 T^{7} - 27417693 T^{8} + 3093430 p T^{9} + 375171 p^{2} T^{10} - 63804 p^{3} T^{11} + 4938 p^{4} T^{12} + 798 p^{5} T^{13} - 89 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 + 24 T + 380 T^{2} + 3936 T^{3} + 34742 T^{4} + 3936 p T^{5} + 380 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 15 T + 297 T^{2} - 2816 T^{3} + 30987 T^{4} - 2816 p T^{5} + 297 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 6 T - 153 T^{2} - 198 T^{3} + 15790 T^{4} - 18060 T^{5} - 946377 T^{6} + 602562 T^{7} + 42720099 T^{8} + 602562 p T^{9} - 946377 p^{2} T^{10} - 18060 p^{3} T^{11} + 15790 p^{4} T^{12} - 198 p^{5} T^{13} - 153 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - T - 284 T^{2} + 143 T^{3} + 50058 T^{4} - 14706 T^{5} - 5753901 T^{6} + 372816 T^{7} + 494276869 T^{8} + 372816 p T^{9} - 5753901 p^{2} T^{10} - 14706 p^{3} T^{11} + 50058 p^{4} T^{12} + 143 p^{5} T^{13} - 284 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 12 T - 95 T^{2} - 1656 T^{3} + 26 p T^{4} + 49164 T^{5} - 718131 T^{6} + 1559838 T^{7} + 122032043 T^{8} + 1559838 p T^{9} - 718131 p^{2} T^{10} + 49164 p^{3} T^{11} + 26 p^{5} T^{12} - 1656 p^{5} T^{13} - 95 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 16 T + 246 T^{2} - 2655 T^{3} + 28977 T^{4} - 2655 p T^{5} + 246 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 25 T + 99 T^{2} - 560 T^{3} + 27488 T^{4} + 325175 T^{5} - 858096 T^{6} + 13373550 T^{7} + 499580157 T^{8} + 13373550 p T^{9} - 858096 p^{2} T^{10} + 325175 p^{3} T^{11} + 27488 p^{4} T^{12} - 560 p^{5} T^{13} + 99 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + T - 199 T^{2} - 634 T^{3} + 12828 T^{4} + 70339 T^{5} - 1453960 T^{6} - 2228292 T^{7} + 235898857 T^{8} - 2228292 p T^{9} - 1453960 p^{2} T^{10} + 70339 p^{3} T^{11} + 12828 p^{4} T^{12} - 634 p^{5} T^{13} - 199 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.95116089530870373609332517127, −3.92449994618610155107089706096, −3.88032408228918847159345202963, −3.78620171509677792321050629097, −3.34844079628850142710156451529, −3.23011744309070769780625940376, −3.22848983568010668701760267079, −3.11750101373568749863973045128, −3.08314149127824459320673045548, −2.78453657187582889643177857828, −2.75399287474828974674227772555, −2.62346605595881490963137445269, −2.62103945657343960517990458527, −2.26429670053348643629428087196, −2.21264890544501868799108124149, −2.13705916819401050220073148928, −2.07126156185774014372992943190, −1.77767944174787153497943066335, −1.44623185140987062625169887764, −1.38036062653787145110953187871, −1.36706633605097688247346449056, −0.894450758319033610378501108969, −0.22141033114732812454576925114, −0.15788906209981625401113486649, −0.10779337996681620842433225529, 0.10779337996681620842433225529, 0.15788906209981625401113486649, 0.22141033114732812454576925114, 0.894450758319033610378501108969, 1.36706633605097688247346449056, 1.38036062653787145110953187871, 1.44623185140987062625169887764, 1.77767944174787153497943066335, 2.07126156185774014372992943190, 2.13705916819401050220073148928, 2.21264890544501868799108124149, 2.26429670053348643629428087196, 2.62103945657343960517990458527, 2.62346605595881490963137445269, 2.75399287474828974674227772555, 2.78453657187582889643177857828, 3.08314149127824459320673045548, 3.11750101373568749863973045128, 3.22848983568010668701760267079, 3.23011744309070769780625940376, 3.34844079628850142710156451529, 3.78620171509677792321050629097, 3.88032408228918847159345202963, 3.92449994618610155107089706096, 3.95116089530870373609332517127
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.