Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 4·4-s + 5-s + 2·6-s − 16·7-s − 6·8-s + 5·9-s − 2·10-s + 4·11-s − 4·12-s − 8·13-s + 32·14-s − 15-s + 11·16-s − 9·17-s − 10·18-s + 4·19-s + 4·20-s + 16·21-s − 8·22-s − 6·23-s + 6·24-s + 13·25-s + 16·26-s − 6·27-s − 64·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 2·4-s + 0.447·5-s + 0.816·6-s − 6.04·7-s − 2.12·8-s + 5/3·9-s − 0.632·10-s + 1.20·11-s − 1.15·12-s − 2.21·13-s + 8.55·14-s − 0.258·15-s + 11/4·16-s − 2.18·17-s − 2.35·18-s + 0.917·19-s + 0.894·20-s + 3.49·21-s − 1.70·22-s − 1.25·23-s + 1.22·24-s + 13/5·25-s + 3.13·26-s − 1.15·27-s − 12.0·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{8} \cdot 11^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.66605\times 10^{7}\) |
| Root analytic conductor: | \(2.82719\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{8} \cdot 11^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2389185746\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2389185746\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} )^{4} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( ( 1 + T - p T^{2} - p T^{3} + 3 T^{4} - p^{2} T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} )( 1 + T + T^{2} + T^{3} - 3 T^{4} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( 1 + T - 4 T^{2} - p T^{3} + 8 T^{4} + 2 p T^{5} + 29 T^{6} + 4 T^{7} - 131 T^{8} + 4 p T^{9} + 29 p^{2} T^{10} + 2 p^{4} T^{11} + 8 p^{4} T^{12} - p^{6} T^{13} - 4 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 - T - 12 T^{2} + 7 T^{3} + 78 T^{4} - 18 T^{5} - 353 T^{6} + 8 p T^{7} + 1461 T^{8} + 8 p^{2} T^{9} - 353 p^{2} T^{10} - 18 p^{3} T^{11} + 78 p^{4} T^{12} + 7 p^{5} T^{13} - 12 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 9 T - 9 T^{2} - 120 T^{3} + 1454 T^{4} + 321 p T^{5} - 24390 T^{6} - 228 T^{7} + 752181 T^{8} - 228 p T^{9} - 24390 p^{2} T^{10} + 321 p^{4} T^{11} + 1454 p^{4} T^{12} - 120 p^{5} T^{13} - 9 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T - 22 T^{2} - 122 T^{3} + 923 T^{4} + 2833 T^{5} + 9003 T^{6} - 67563 T^{7} - 223871 T^{8} - 67563 p T^{9} + 9003 p^{2} T^{10} + 2833 p^{3} T^{11} + 923 p^{4} T^{12} - 122 p^{5} T^{13} - 22 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T - 51 T^{2} - 198 T^{3} + 2492 T^{4} + 4596 T^{5} - 81813 T^{6} - 41460 T^{7} + 2120271 T^{8} - 41460 p T^{9} - 81813 p^{2} T^{10} + 4596 p^{3} T^{11} + 2492 p^{4} T^{12} - 198 p^{5} T^{13} - 51 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 + 15 T + 192 T^{2} + 1455 T^{3} + 329 p T^{4} + 1455 p T^{5} + 192 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 - 10 T + 36 T^{2} - 80 T^{3} + 263 T^{4} - 5940 T^{5} + 48496 T^{6} - 91150 T^{7} - 264888 T^{8} - 91150 p T^{9} + 48496 p^{2} T^{10} - 5940 p^{3} T^{11} + 263 p^{4} T^{12} - 80 p^{5} T^{13} + 36 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 9 T - 47 T^{2} - 402 T^{3} + 2968 T^{4} + 12591 T^{5} - 126692 T^{6} - 9084 p T^{7} + 2818885 T^{8} - 9084 p^{2} T^{9} - 126692 p^{2} T^{10} + 12591 p^{3} T^{11} + 2968 p^{4} T^{12} - 402 p^{5} T^{13} - 47 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 + 5 T + 157 T^{2} + 614 T^{3} + 9513 T^{4} + 614 p T^{5} + 157 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 7 T + 88 T^{2} - 151 T^{3} + 2479 T^{4} - 151 p T^{5} + 88 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 2 T - 177 T^{2} + 194 T^{3} + 19602 T^{4} - 276 p T^{5} - 1430333 T^{6} + 218954 T^{7} + 78785115 T^{8} + 218954 p T^{9} - 1430333 p^{2} T^{10} - 276 p^{4} T^{11} + 19602 p^{4} T^{12} + 194 p^{5} T^{13} - 177 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 27 T + 324 T^{2} - 2775 T^{3} + 20984 T^{4} - 93066 T^{5} - 233595 T^{6} + 6263826 T^{7} - 52673079 T^{8} + 6263826 p T^{9} - 233595 p^{2} T^{10} - 93066 p^{3} T^{11} + 20984 p^{4} T^{12} - 2775 p^{5} T^{13} + 324 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 4 T - 132 T^{2} + 56 T^{3} + 10218 T^{4} - 29436 T^{5} - 531920 T^{6} + 1046468 T^{7} + 26626323 T^{8} + 1046468 p T^{9} - 531920 p^{2} T^{10} - 29436 p^{3} T^{11} + 10218 p^{4} T^{12} + 56 p^{5} T^{13} - 132 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 19 T + 17 T^{2} - 502 T^{3} + 17372 T^{4} + 129281 T^{5} - 866544 T^{6} + 956160 T^{7} + 117335773 T^{8} + 956160 p T^{9} - 866544 p^{2} T^{10} + 129281 p^{3} T^{11} + 17372 p^{4} T^{12} - 502 p^{5} T^{13} + 17 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - T - 157 T^{2} + 724 T^{3} + 12050 T^{4} - 74279 T^{5} - 387276 T^{6} + 2826234 T^{7} + 10932511 T^{8} + 2826234 p T^{9} - 387276 p^{2} T^{10} - 74279 p^{3} T^{11} + 12050 p^{4} T^{12} + 724 p^{5} T^{13} - 157 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 5 T + 109 T^{2} + 440 T^{3} + 6021 T^{4} + 440 p T^{5} + 109 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 + 2 T - 181 T^{2} + 310 T^{3} + 18122 T^{4} - 56432 T^{5} - 999705 T^{6} + 2420754 T^{7} + 49641019 T^{8} + 2420754 p T^{9} - 999705 p^{2} T^{10} - 56432 p^{3} T^{11} + 18122 p^{4} T^{12} + 310 p^{5} T^{13} - 181 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 32 T + 354 T^{2} - 3154 T^{3} + 60653 T^{4} - 725163 T^{5} + 4744315 T^{6} - 56426261 T^{7} + 718187661 T^{8} - 56426261 p T^{9} + 4744315 p^{2} T^{10} - 725163 p^{3} T^{11} + 60653 p^{4} T^{12} - 3154 p^{5} T^{13} + 354 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 86 T^{2} - 243 T^{3} + 11919 T^{4} - 243 p T^{5} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 3 T - 141 T^{2} - 1380 T^{3} + 13778 T^{4} + 171837 T^{5} + 1199130 T^{6} - 13390452 T^{7} - 145165635 T^{8} - 13390452 p T^{9} + 1199130 p^{2} T^{10} + 171837 p^{3} T^{11} + 13778 p^{4} T^{12} - 1380 p^{5} T^{13} - 141 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 3 T + 341 T^{2} - 840 T^{3} + 47349 T^{4} - 840 p T^{5} + 341 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−4.21547836448463679222022634320, −3.96809401874888046707382831609, −3.84679141306469630971877023886, −3.82042591414134241989307281571, −3.78378376498182398456180682425, −3.54673975595993602659377974279, −3.34956821602600283514152822234, −3.33239020998424179710683649760, −3.19015284366301158297670803538, −3.15795965127436589787455842325, −2.77972782441176388148755721982, −2.77896969223186263240362503888, −2.69965535637817066070675472200, −2.52200936388216359750991216008, −2.09376378478304832199895717832, −2.05912350808478815415480649097, −2.03272947816900124217668441931, −1.90616597396439900450387932973, −1.87369625632118614688571595851, −1.22471206379033969766324618253, −1.19865833515620129766655747683, −0.72924863423106269620514761795, −0.64226233258619270938285292397, −0.40288710303739366681501570728, −0.17257459404037613051318386196, 0.17257459404037613051318386196, 0.40288710303739366681501570728, 0.64226233258619270938285292397, 0.72924863423106269620514761795, 1.19865833515620129766655747683, 1.22471206379033969766324618253, 1.87369625632118614688571595851, 1.90616597396439900450387932973, 2.03272947816900124217668441931, 2.05912350808478815415480649097, 2.09376378478304832199895717832, 2.52200936388216359750991216008, 2.69965535637817066070675472200, 2.77896969223186263240362503888, 2.77972782441176388148755721982, 3.15795965127436589787455842325, 3.19015284366301158297670803538, 3.33239020998424179710683649760, 3.34956821602600283514152822234, 3.54673975595993602659377974279, 3.78378376498182398456180682425, 3.82042591414134241989307281571, 3.84679141306469630971877023886, 3.96809401874888046707382831609, 4.21547836448463679222022634320
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.