Dirichlet series
| L(s) = 1 | − 8·2-s + 36·4-s − 2·5-s + 3·7-s − 120·8-s + 16·10-s − 4·11-s + 3·13-s − 24·14-s + 330·16-s − 4·17-s − 4·19-s − 72·20-s + 32·22-s + 8·23-s + 13·25-s − 24·26-s + 108·28-s − 2·29-s + 14·31-s − 792·32-s + 32·34-s − 6·35-s + 12·37-s + 32·38-s + 240·40-s − 12·41-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 0.894·5-s + 1.13·7-s − 42.4·8-s + 5.05·10-s − 1.20·11-s + 0.832·13-s − 6.41·14-s + 82.5·16-s − 0.970·17-s − 0.917·19-s − 16.0·20-s + 6.82·22-s + 1.66·23-s + 13/5·25-s − 4.70·26-s + 20.4·28-s − 0.371·29-s + 2.51·31-s − 140.·32-s + 5.48·34-s − 1.01·35-s + 1.97·37-s + 5.19·38-s + 37.9·40-s − 1.87·41-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
| \(L(1)\) | \(\approx\) | \(0.07996863247\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07996863247\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 3 T + 2 T^{2} + 3 p T^{3} - 117 T^{4} + 3 p^{2} T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 - 3 T + 20 T^{2} - 15 T^{3} + 231 T^{4} - 15 p T^{5} + 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( ( 1 - p T + 4 T^{2} + 19 T^{3} - 63 T^{4} + 19 p T^{5} + 4 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + 7 T + 22 T^{2} + 37 T^{3} + 57 T^{4} + 37 p T^{5} + 22 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 + 4 T - 14 T^{2} - 162 T^{3} - 81 T^{4} + 2445 T^{5} + 8427 T^{6} - 14419 T^{7} - 125751 T^{8} - 14419 p T^{9} + 8427 p^{2} T^{10} + 2445 p^{3} T^{11} - 81 p^{4} T^{12} - 162 p^{5} T^{13} - 14 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 + 2 T + 24 T^{2} - 3 p T^{3} + 211 T^{4} - 3 p^{2} T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( 1 + 4 T - p T^{2} + 104 T^{3} + 698 T^{4} - 2668 T^{5} + 8301 T^{6} + 62226 T^{7} - 152189 T^{8} + 62226 p T^{9} + 8301 p^{2} T^{10} - 2668 p^{3} T^{11} + 698 p^{4} T^{12} + 104 p^{5} T^{13} - p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 4 T + 58 T^{2} - 195 T^{3} + 1613 T^{4} - 195 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 2 T - 20 T^{2} + 162 T^{3} - 79 T^{4} - 6097 T^{5} + 28467 T^{6} + 124657 T^{7} - 499445 T^{8} + 124657 p T^{9} + 28467 p^{2} T^{10} - 6097 p^{3} T^{11} - 79 p^{4} T^{12} + 162 p^{5} T^{13} - 20 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 14 T + 44 T^{2} + 2 p T^{3} + 741 T^{4} - 3461 T^{5} - 53077 T^{6} + 156063 T^{7} + 844921 T^{8} + 156063 p T^{9} - 53077 p^{2} T^{10} - 3461 p^{3} T^{11} + 741 p^{4} T^{12} + 2 p^{6} T^{13} + 44 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 6 T + 156 T^{2} - 665 T^{3} + 8805 T^{4} - 665 p T^{5} + 156 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 12 T - 6 T^{2} - 306 T^{3} + 1865 T^{4} + 9147 T^{5} - 104817 T^{6} - 676101 T^{7} - 2276181 T^{8} - 676101 p T^{9} - 104817 p^{2} T^{10} + 9147 p^{3} T^{11} + 1865 p^{4} T^{12} - 306 p^{5} T^{13} - 6 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 138 T^{2} + 146 T^{3} + 10887 T^{4} - 13213 T^{5} - 610013 T^{6} + 270903 T^{7} + 27676345 T^{8} + 270903 p T^{9} - 610013 p^{2} T^{10} - 13213 p^{3} T^{11} + 10887 p^{4} T^{12} + 146 p^{5} T^{13} - 138 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 + 7 T - 107 T^{2} - 654 T^{3} + 8042 T^{4} + 34657 T^{5} - 424428 T^{6} - 772924 T^{7} + 19410175 T^{8} - 772924 p T^{9} - 424428 p^{2} T^{10} + 34657 p^{3} T^{11} + 8042 p^{4} T^{12} - 654 p^{5} T^{13} - 107 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - T - 65 T^{2} + 1080 T^{3} + 894 T^{4} - 62409 T^{5} + 478074 T^{6} + 2019040 T^{7} - 28883847 T^{8} + 2019040 p T^{9} + 478074 p^{2} T^{10} - 62409 p^{3} T^{11} + 894 p^{4} T^{12} + 1080 p^{5} T^{13} - 65 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( ( 1 - 16 T + 253 T^{2} - 2356 T^{3} + 21813 T^{4} - 2356 p T^{5} + 253 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 + 4 T - 52 T^{2} + 1256 T^{3} + 6170 T^{4} - 66844 T^{5} + 839472 T^{6} + 5794044 T^{7} - 38376173 T^{8} + 5794044 p T^{9} + 839472 p^{2} T^{10} - 66844 p^{3} T^{11} + 6170 p^{4} T^{12} + 1256 p^{5} T^{13} - 52 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 19 T + 76 T^{2} + 197 T^{3} + 2400 T^{4} - 18294 T^{5} - 410307 T^{6} + 5554842 T^{7} - 44341751 T^{8} + 5554842 p T^{9} - 410307 p^{2} T^{10} - 18294 p^{3} T^{11} + 2400 p^{4} T^{12} + 197 p^{5} T^{13} + 76 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 20 T + 151 T^{2} + 948 T^{3} + 5336 T^{4} - 8680 T^{5} - 357951 T^{6} - 6067808 T^{7} - 73033073 T^{8} - 6067808 p T^{9} - 357951 p^{2} T^{10} - 8680 p^{3} T^{11} + 5336 p^{4} T^{12} + 948 p^{5} T^{13} + 151 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 7 T - 176 T^{2} - 557 T^{3} + 21594 T^{4} + 15702 T^{5} - 1863345 T^{6} - 909252 T^{7} + 122212573 T^{8} - 909252 p T^{9} - 1863345 p^{2} T^{10} + 15702 p^{3} T^{11} + 21594 p^{4} T^{12} - 557 p^{5} T^{13} - 176 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 189 T^{2} - 124 T^{3} - 6786 T^{4} + 100964 T^{5} - 891383 T^{6} - 39114 p T^{7} + 106087891 T^{8} - 39114 p^{2} T^{9} - 891383 p^{2} T^{10} + 100964 p^{3} T^{11} - 6786 p^{4} T^{12} - 124 p^{5} T^{13} + 189 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 32 T + 696 T^{2} - 9651 T^{3} + 104187 T^{4} - 9651 p T^{5} + 696 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( ( 1 + 11 T + 252 T^{2} + 2433 T^{3} + 31963 T^{4} + 2433 p T^{5} + 252 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 11 T - 187 T^{2} + 962 T^{3} + 32064 T^{4} - 25625 T^{5} - 3767656 T^{6} + 3490380 T^{7} + 292157209 T^{8} + 3490380 p T^{9} - 3767656 p^{2} T^{10} - 25625 p^{3} T^{11} + 32064 p^{4} T^{12} + 962 p^{5} T^{13} - 187 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.96147254319672791967769607895, −3.56655260769415483596703925418, −3.52869125071885916129857165286, −3.32121652258163894092631306555, −3.29110670230818779875389013050, −3.27016090880937272270066531023, −3.17541375850223219713077985092, −2.87810972838036403759172989540, −2.79131447440192711376642739527, −2.63901269997120126674357619257, −2.30314115480188407843592789313, −2.27479833596973236567363814654, −2.23965420905423160985402510918, −2.20038616441397662251642543388, −1.96610405212309024831764852596, −1.93960397986032415393729693512, −1.83454194351180220698543867826, −1.27649455352721273658798512533, −1.18513438817106524617286437141, −1.10861780422736724208706849367, −0.823384135428319613846666851292, −0.822645018586192293473634654469, −0.807562358209331671621075562810, −0.53723574596187774795764734625, −0.093047059849126789892231493337, 0.093047059849126789892231493337, 0.53723574596187774795764734625, 0.807562358209331671621075562810, 0.822645018586192293473634654469, 0.823384135428319613846666851292, 1.10861780422736724208706849367, 1.18513438817106524617286437141, 1.27649455352721273658798512533, 1.83454194351180220698543867826, 1.93960397986032415393729693512, 1.96610405212309024831764852596, 2.20038616441397662251642543388, 2.23965420905423160985402510918, 2.27479833596973236567363814654, 2.30314115480188407843592789313, 2.63901269997120126674357619257, 2.79131447440192711376642739527, 2.87810972838036403759172989540, 3.17541375850223219713077985092, 3.27016090880937272270066531023, 3.29110670230818779875389013050, 3.32121652258163894092631306555, 3.52869125071885916129857165286, 3.56655260769415483596703925418, 3.96147254319672791967769607895
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.