Dirichlet series
| L(s) = 1 | − 2-s − 4·3-s + 4·4-s + 3·5-s + 4·6-s + 2·7-s − 2·8-s + 10·9-s − 3·10-s + 5·11-s − 16·12-s + 5·13-s − 2·14-s − 12·15-s + 9·16-s − 11·17-s − 10·18-s − 9·19-s + 12·20-s − 8·21-s − 5·22-s − 16·23-s + 8·24-s + 12·25-s − 5·26-s − 32·27-s + 8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 2·4-s + 1.34·5-s + 1.63·6-s + 0.755·7-s − 0.707·8-s + 10/3·9-s − 0.948·10-s + 1.50·11-s − 4.61·12-s + 1.38·13-s − 0.534·14-s − 3.09·15-s + 9/4·16-s − 2.66·17-s − 2.35·18-s − 2.06·19-s + 2.68·20-s − 1.74·21-s − 1.06·22-s − 3.33·23-s + 1.63·24-s + 12/5·25-s − 0.980·26-s − 6.15·27-s + 1.51·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0204240\) |
| Root analytic conductor: | \(0.784122\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4072448261\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4072448261\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 - 5 T + 26 T^{2} - 75 T^{3} + 251 T^{4} - 75 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 + T - 3 T^{2} - 5 T^{3} + 19 T^{5} + 21 T^{6} - 5 p^{2} T^{7} - 51 T^{8} - 5 p^{3} T^{9} + 21 p^{2} T^{10} + 19 p^{3} T^{11} - 5 p^{5} T^{13} - 3 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 + 2 T + T^{2} + 2 p T^{3} + 19 T^{4} + 2 p^{2} T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 5 | \( 1 - 3 T - 3 T^{2} + 6 p T^{3} - 39 T^{4} - 6 p^{2} T^{5} + 428 T^{6} + 231 T^{7} - 2169 T^{8} + 231 p T^{9} + 428 p^{2} T^{10} - 6 p^{5} T^{11} - 39 p^{4} T^{12} + 6 p^{6} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 35 T^{2} - 90 T^{3} + 47 p T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 8865 p T^{9} + 4380 p^{2} T^{10} - 670 p^{3} T^{11} + 47 p^{5} T^{12} - 90 p^{5} T^{13} + 35 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 11 T + 39 T^{2} + 139 T^{3} + 1249 T^{4} + 7484 T^{5} + 34098 T^{6} + 128922 T^{7} + 457683 T^{8} + 128922 p T^{9} + 34098 p^{2} T^{10} + 7484 p^{3} T^{11} + 1249 p^{4} T^{12} + 139 p^{5} T^{13} + 39 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 86580 p T^{9} - 12940 p^{2} T^{10} + 1674 p^{3} T^{11} + 1023 p^{4} T^{12} + 9 p^{6} T^{13} + 43 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 402 p T^{5} + 83 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 191700 p T^{9} + 81940 p^{2} T^{10} + 11754 p^{3} T^{11} - 762 p^{4} T^{12} - 429 p^{5} T^{13} - 22 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 25700 p T^{9} - 31430 p^{2} T^{10} + 12836 p^{3} T^{11} + 3210 p^{4} T^{12} + 183 p^{5} T^{13} + 34 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 92390 p T^{9} - 20209 p^{2} T^{10} - 12954 p^{3} T^{11} + 1350 p^{4} T^{12} + 520 p^{5} T^{13} - 63 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 22 T + 160 T^{2} + 414 T^{3} + 2641 T^{4} + 46342 T^{5} + 334288 T^{6} + 25916 p T^{7} + 2274033 T^{8} + 25916 p^{2} T^{9} + 334288 p^{2} T^{10} + 46342 p^{3} T^{11} + 2641 p^{4} T^{12} + 414 p^{5} T^{13} + 160 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 47 | \( 1 - 7 T - 100 T^{2} + 1414 T^{3} + 286 T^{4} - 103523 T^{5} + 579636 T^{6} + 2543800 T^{7} - 42803781 T^{8} + 2543800 p T^{9} + 579636 p^{2} T^{10} - 103523 p^{3} T^{11} + 286 p^{4} T^{12} + 1414 p^{5} T^{13} - 100 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T - 173 T^{2} + 964 T^{3} + 9936 T^{4} - 104742 T^{5} + 122655 T^{6} + 3229712 T^{7} - 31796061 T^{8} + 3229712 p T^{9} + 122655 p^{2} T^{10} - 104742 p^{3} T^{11} + 9936 p^{4} T^{12} + 964 p^{5} T^{13} - 173 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 25 T + 206 T^{2} + 25 T^{3} - 15220 T^{4} + 157300 T^{5} - 429276 T^{6} - 8492650 T^{7} + 113408229 T^{8} - 8492650 p T^{9} - 429276 p^{2} T^{10} + 157300 p^{3} T^{11} - 15220 p^{4} T^{12} + 25 p^{5} T^{13} + 206 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 5279480 p T^{9} + 1092660 p^{2} T^{10} - 164342 p^{3} T^{11} + 318 p^{4} T^{12} + 1783 p^{5} T^{13} - 108 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 4047848 p T^{9} + 728307 p^{2} T^{10} + 52059 p^{3} T^{11} + 6927 p^{4} T^{12} - 147 p^{5} T^{13} - 9 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 13580 p T^{9} + 620974 p^{2} T^{10} - 100292 p^{3} T^{11} + 220 p^{4} T^{12} - 345 p^{5} T^{13} - 2 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 1473567 p T^{9} - 1553178 p^{2} T^{10} + 24564 p^{3} T^{11} + 10031 p^{4} T^{12} - 768 p^{5} T^{13} - 75 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 23 T + 98 T^{2} + 1745 T^{3} - 15470 T^{4} - 98632 T^{5} + 1750974 T^{6} + 1504700 T^{7} - 137136651 T^{8} + 1504700 p T^{9} + 1750974 p^{2} T^{10} - 98632 p^{3} T^{11} - 15470 p^{4} T^{12} + 1745 p^{5} T^{13} + 98 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 3419 p T^{5} + 312 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 32754150 p T^{9} - 577417 p^{2} T^{10} - 83370 p^{3} T^{11} + 10062 p^{4} T^{12} - 2160 p^{5} T^{13} + 361 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.97121161412748877205552351327, −6.58658744842989403185398835552, −6.55853959900630711951262498520, −6.33950326340247447495055540632, −6.18815130029869552076252360432, −6.05516119523766669621805520064, −5.91897077543004412185023096152, −5.91440232868672542697155407730, −5.68499769716732606148163606524, −5.47522674241852729326163910318, −5.22570926745566645016131740109, −5.01508108947484100816107803017, −4.86459117125640729408291066182, −4.28011407875480074429386371100, −4.19664684850363922255548477027, −4.09184198463952957163798532881, −4.01998150583811817044212011463, −3.77869344625928192220878576802, −3.48409995540354735534011642463, −2.98273854383731678286599464081, −2.31691111158093558122197604300, −2.00084791067807242737199077908, −1.85476106025871693512891186687, −1.83225717415857255392004335705, −1.66189493881005694540507173399, 1.66189493881005694540507173399, 1.83225717415857255392004335705, 1.85476106025871693512891186687, 2.00084791067807242737199077908, 2.31691111158093558122197604300, 2.98273854383731678286599464081, 3.48409995540354735534011642463, 3.77869344625928192220878576802, 4.01998150583811817044212011463, 4.09184198463952957163798532881, 4.19664684850363922255548477027, 4.28011407875480074429386371100, 4.86459117125640729408291066182, 5.01508108947484100816107803017, 5.22570926745566645016131740109, 5.47522674241852729326163910318, 5.68499769716732606148163606524, 5.91440232868672542697155407730, 5.91897077543004412185023096152, 6.05516119523766669621805520064, 6.18815130029869552076252360432, 6.33950326340247447495055540632, 6.55853959900630711951262498520, 6.58658744842989403185398835552, 6.97121161412748877205552351327
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.