Dirichlet series
| L(s) = 1 | + 2·2-s − 3-s + 4-s + 5·5-s − 2·6-s − 2·7-s + 10·10-s − 5·11-s − 12-s + 3·13-s − 4·14-s − 5·15-s − 7·17-s − 14·19-s + 5·20-s + 2·21-s − 10·22-s − 20·23-s + 5·25-s + 6·26-s + 7·27-s − 2·28-s − 23·29-s − 10·30-s + 3·31-s − 2·32-s + 5·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 1/2·4-s + 2.23·5-s − 0.816·6-s − 0.755·7-s + 3.16·10-s − 1.50·11-s − 0.288·12-s + 0.832·13-s − 1.06·14-s − 1.29·15-s − 1.69·17-s − 3.21·19-s + 1.11·20-s + 0.436·21-s − 2.13·22-s − 4.17·23-s + 25-s + 1.17·26-s + 1.34·27-s − 0.377·28-s − 4.27·29-s − 1.82·30-s + 0.538·31-s − 0.353·32-s + 0.870·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.22856\) |
| Root analytic conductor: | \(1.10891\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.021103284\) |
| \(L(\frac12)\) | \(\approx\) | \(1.021103284\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 + 5 T + 9 T^{2} + 65 T^{3} + 356 T^{4} + 65 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + T + T^{2} - 2 p T^{3} - 22 T^{4} - 13 p T^{5} + 4 T^{6} + 34 p T^{7} + 313 T^{8} + 34 p^{2} T^{9} + 4 p^{2} T^{10} - 13 p^{4} T^{11} - 22 p^{4} T^{12} - 2 p^{6} T^{13} + p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 - p T + 3 p T^{2} - 7 p T^{3} + 16 p T^{4} - 7 p^{2} T^{5} + 3 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + p T^{2} - 4 p T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8} ) \) | |
| 13 | \( 1 - 3 T + 27 T^{2} - 74 T^{3} + 306 T^{4} - 333 T^{5} + 830 T^{6} + 8288 T^{7} - 17351 T^{8} + 8288 p T^{9} + 830 p^{2} T^{10} - 333 p^{3} T^{11} + 306 p^{4} T^{12} - 74 p^{5} T^{13} + 27 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 7 T - 21 T^{2} - 188 T^{3} + 168 T^{4} + 3787 T^{5} + 18786 T^{6} - 12494 T^{7} - 433107 T^{8} - 12494 p T^{9} + 18786 p^{2} T^{10} + 3787 p^{3} T^{11} + 168 p^{4} T^{12} - 188 p^{5} T^{13} - 21 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 14 T + 44 T^{2} - 391 T^{3} - 3289 T^{4} - 3278 T^{5} + 43461 T^{6} + 106567 T^{7} - 132138 T^{8} + 106567 p T^{9} + 43461 p^{2} T^{10} - 3278 p^{3} T^{11} - 3289 p^{4} T^{12} - 391 p^{5} T^{13} + 44 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 10 T + 117 T^{2} + 30 p T^{3} + 4304 T^{4} + 30 p^{2} T^{5} + 117 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 23 T + 215 T^{2} + 1150 T^{3} + 4920 T^{4} + 16319 T^{5} - 588 T^{6} - 218840 T^{7} - 941275 T^{8} - 218840 p T^{9} - 588 p^{2} T^{10} + 16319 p^{3} T^{11} + 4920 p^{4} T^{12} + 1150 p^{5} T^{13} + 215 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 3 T - 59 T^{2} + 288 T^{3} + 1446 T^{4} - 7059 T^{5} - 13756 T^{6} + 77634 T^{7} + 295787 T^{8} + 77634 p T^{9} - 13756 p^{2} T^{10} - 7059 p^{3} T^{11} + 1446 p^{4} T^{12} + 288 p^{5} T^{13} - 59 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T + 28 T^{2} + 108 T^{3} - 1659 T^{4} + 15654 T^{5} - 7960 T^{6} - 302976 T^{7} + 3110369 T^{8} - 302976 p T^{9} - 7960 p^{2} T^{10} + 15654 p^{3} T^{11} - 1659 p^{4} T^{12} + 108 p^{5} T^{13} + 28 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 2 T - 34 T^{2} - 22 T^{3} + 3431 T^{4} - 14 T^{5} - 202656 T^{6} - 215616 T^{7} + 5961697 T^{8} - 215616 p T^{9} - 202656 p^{2} T^{10} - 14 p^{3} T^{11} + 3431 p^{4} T^{12} - 22 p^{5} T^{13} - 34 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 8 T + 151 T^{2} + 874 T^{3} + 9259 T^{4} + 874 p T^{5} + 151 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 34 T + 558 T^{2} - 6218 T^{3} + 53101 T^{4} - 339514 T^{5} + 1482870 T^{6} - 3941964 T^{7} + 10000719 T^{8} - 3941964 p T^{9} + 1482870 p^{2} T^{10} - 339514 p^{3} T^{11} + 53101 p^{4} T^{12} - 6218 p^{5} T^{13} + 558 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 13 T + 57 T^{2} - 364 T^{3} + 7446 T^{4} - 24973 T^{5} - 308520 T^{6} + 1610408 T^{7} + 4923609 T^{8} + 1610408 p T^{9} - 308520 p^{2} T^{10} - 24973 p^{3} T^{11} + 7446 p^{4} T^{12} - 364 p^{5} T^{13} + 57 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 3 T + 70 T^{2} - 290 T^{3} + 90 T^{4} - 43981 T^{5} + 204307 T^{6} + 662240 T^{7} + 31216160 T^{8} + 662240 p T^{9} + 204307 p^{2} T^{10} - 43981 p^{3} T^{11} + 90 p^{4} T^{12} - 290 p^{5} T^{13} + 70 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 16 T + 60 T^{2} - 700 T^{3} - 8915 T^{4} - 17512 T^{5} + 292088 T^{6} + 2111920 T^{7} + 8729225 T^{8} + 2111920 p T^{9} + 292088 p^{2} T^{10} - 17512 p^{3} T^{11} - 8915 p^{4} T^{12} - 700 p^{5} T^{13} + 60 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 1698 p T^{5} + 224 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 45 T + 833 T^{2} - 7690 T^{3} + 23498 T^{4} + 282055 T^{5} - 3945394 T^{6} + 19376700 T^{7} - 63081145 T^{8} + 19376700 p T^{9} - 3945394 p^{2} T^{10} + 282055 p^{3} T^{11} + 23498 p^{4} T^{12} - 7690 p^{5} T^{13} + 833 p^{6} T^{14} - 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - T - 4 T^{2} - 604 T^{3} - 2692 T^{4} + 66769 T^{5} + 162519 T^{6} - 1285592 T^{7} - 48126792 T^{8} - 1285592 p T^{9} + 162519 p^{2} T^{10} + 66769 p^{3} T^{11} - 2692 p^{4} T^{12} - 604 p^{5} T^{13} - 4 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 21 T + 9 T^{2} - 3884 T^{3} - 29964 T^{4} + 277543 T^{5} + 4381786 T^{6} - 7795662 T^{7} - 390660113 T^{8} - 7795662 p T^{9} + 4381786 p^{2} T^{10} + 277543 p^{3} T^{11} - 29964 p^{4} T^{12} - 3884 p^{5} T^{13} + 9 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 12 T + 147 T^{2} - 546 T^{3} + 11481 T^{4} + 46458 T^{5} - 690295 T^{6} + 17986632 T^{7} - 78881196 T^{8} + 17986632 p T^{9} - 690295 p^{2} T^{10} + 46458 p^{3} T^{11} + 11481 p^{4} T^{12} - 546 p^{5} T^{13} + 147 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 1158 p T^{5} + 297 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 5 T - 124 T^{2} - 520 T^{3} + 7232 T^{4} + 142605 T^{5} + 1287703 T^{6} - 12152100 T^{7} - 156566960 T^{8} - 12152100 p T^{9} + 1287703 p^{2} T^{10} + 142605 p^{3} T^{11} + 7232 p^{4} T^{12} - 520 p^{5} T^{13} - 124 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.89762457389607825250173926974, −5.88463115902114304932964297463, −5.77523366909982989092103493895, −5.41751781824761196291114355034, −5.38024899945685105466735002249, −5.33968262747354690819818429042, −5.17203543773667181764765487560, −4.75718926117945437472167195868, −4.74198108150658984673783147816, −4.56517503748062788273818367528, −4.13362120239232191092331892054, −4.09318858661731543724910863366, −3.97800001387445315890539175838, −3.85190251637407995071700714916, −3.71663529730624689214710375134, −3.65897396694903842007938837651, −3.20380010543030768510856674837, −2.77566173915414623639593032887, −2.43100247566667216656776376366, −2.26401140672972347514027091273, −2.22846785621780543874870388604, −2.10880567635175124580303990422, −1.97429048964656077914805595663, −1.59886887730278695533608751329, −0.37683172127169970493044285486, 0.37683172127169970493044285486, 1.59886887730278695533608751329, 1.97429048964656077914805595663, 2.10880567635175124580303990422, 2.22846785621780543874870388604, 2.26401140672972347514027091273, 2.43100247566667216656776376366, 2.77566173915414623639593032887, 3.20380010543030768510856674837, 3.65897396694903842007938837651, 3.71663529730624689214710375134, 3.85190251637407995071700714916, 3.97800001387445315890539175838, 4.09318858661731543724910863366, 4.13362120239232191092331892054, 4.56517503748062788273818367528, 4.74198108150658984673783147816, 4.75718926117945437472167195868, 5.17203543773667181764765487560, 5.33968262747354690819818429042, 5.38024899945685105466735002249, 5.41751781824761196291114355034, 5.77523366909982989092103493895, 5.88463115902114304932964297463, 5.89762457389607825250173926974
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.