Dirichlet series
| L(s) = 1 | + 4·2-s − 2·3-s + 11·4-s − 2·5-s − 8·6-s + 3·7-s + 23·8-s + 9-s − 8·10-s + 3·11-s − 22·12-s − 4·13-s + 12·14-s + 4·15-s + 39·16-s + 4·18-s + 2·19-s − 22·20-s − 6·21-s + 12·22-s − 6·23-s − 46·24-s + 25-s − 16·26-s + 33·28-s + 10·29-s + 16·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 11/2·4-s − 0.894·5-s − 3.26·6-s + 1.13·7-s + 8.13·8-s + 1/3·9-s − 2.52·10-s + 0.904·11-s − 6.35·12-s − 1.10·13-s + 3.20·14-s + 1.03·15-s + 39/4·16-s + 0.942·18-s + 0.458·19-s − 4.91·20-s − 1.30·21-s + 2.55·22-s − 1.25·23-s − 9.38·24-s + 1/5·25-s − 3.13·26-s + 6.23·28-s + 1.85·29-s + 2.92·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.08005\) |
| Root analytic conductor: | \(1.14783\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.933282313\) |
| \(L(\frac12)\) | \(\approx\) | \(7.933282313\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 3 T - 2 p T^{2} + 19 T^{3} + 335 T^{4} + 19 p T^{5} - 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 - p^{2} T + 5 T^{2} + T^{3} - 3 p T^{4} - 7 T^{5} + 21 T^{6} + 7 T^{7} - 49 T^{8} + 7 p T^{9} + 21 p^{2} T^{10} - 7 p^{3} T^{11} - 3 p^{5} T^{12} + p^{5} T^{13} + 5 p^{6} T^{14} - p^{9} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 3 T - 3 T^{2} + 24 T^{3} + 3 T^{4} - 24 T^{5} - 206 T^{6} - 573 T^{7} + 5433 T^{8} - 573 p T^{9} - 206 p^{2} T^{10} - 24 p^{3} T^{11} + 3 p^{4} T^{12} + 24 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 4 T + 3 T^{2} + 72 T^{3} + 568 T^{4} + 124 p T^{5} + 2841 T^{6} + 23776 T^{7} + 137583 T^{8} + 23776 p T^{9} + 2841 p^{2} T^{10} + 124 p^{4} T^{11} + 568 p^{4} T^{12} + 72 p^{5} T^{13} + 3 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 13 T^{2} - 80 T^{3} + 270 T^{4} - 460 T^{5} + 5773 T^{6} - 14470 T^{7} + 23699 T^{8} - 14470 p T^{9} + 5773 p^{2} T^{10} - 460 p^{3} T^{11} + 270 p^{4} T^{12} - 80 p^{5} T^{13} + 13 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 - 2 T - 13 T^{2} + 72 T^{3} + 392 T^{4} + 110 T^{5} - 8697 T^{6} - 8904 T^{7} + 339075 T^{8} - 8904 p T^{9} - 8697 p^{2} T^{10} + 110 p^{3} T^{11} + 392 p^{4} T^{12} + 72 p^{5} T^{13} - 13 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 3 T + 37 T^{2} + 78 T^{3} + 1093 T^{4} + 78 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 10 T + 55 T^{2} - 400 T^{3} + 3024 T^{4} - 24690 T^{5} + 5055 p T^{6} - 731720 T^{7} + 4160811 T^{8} - 731720 p T^{9} + 5055 p^{3} T^{10} - 24690 p^{3} T^{11} + 3024 p^{4} T^{12} - 400 p^{5} T^{13} + 55 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 19 T + 173 T^{2} - 1079 T^{3} + 6207 T^{4} - 39850 T^{5} + 231342 T^{6} - 1019972 T^{7} + 4555345 T^{8} - 1019972 p T^{9} + 231342 p^{2} T^{10} - 39850 p^{3} T^{11} + 6207 p^{4} T^{12} - 1079 p^{5} T^{13} + 173 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T - p T^{2} - 83 T^{3} - 989 T^{4} - 8474 T^{5} + 32080 T^{6} + 331596 T^{7} + 1075119 T^{8} + 331596 p T^{9} + 32080 p^{2} T^{10} - 8474 p^{3} T^{11} - 989 p^{4} T^{12} - 83 p^{5} T^{13} - p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 9 T + 3 p T^{2} + 974 T^{3} + 11067 T^{4} + 80950 T^{5} + 664532 T^{6} + 106557 p T^{7} + 769255 p T^{8} + 106557 p^{2} T^{9} + 664532 p^{2} T^{10} + 80950 p^{3} T^{11} + 11067 p^{4} T^{12} + 974 p^{5} T^{13} + 3 p^{7} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 62 T^{2} + 375 T^{3} + 1909 T^{4} + 375 p T^{5} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 19 T + 161 T^{2} + 952 T^{3} + 5681 T^{4} + 21302 T^{5} - 48990 T^{6} - 1137555 T^{7} - 8501903 T^{8} - 1137555 p T^{9} - 48990 p^{2} T^{10} + 21302 p^{3} T^{11} + 5681 p^{4} T^{12} + 952 p^{5} T^{13} + 161 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 25 T + 326 T^{2} - 3620 T^{3} + 38702 T^{4} - 373145 T^{5} + 3256788 T^{6} - 26526610 T^{7} + 201208695 T^{8} - 26526610 p T^{9} + 3256788 p^{2} T^{10} - 373145 p^{3} T^{11} + 38702 p^{4} T^{12} - 3620 p^{5} T^{13} + 326 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 13 T + 47 T^{2} - 47 T^{3} + 1017 T^{4} - 9820 T^{5} + 167838 T^{6} - 2814986 T^{7} + 28479065 T^{8} - 2814986 p T^{9} + 167838 p^{2} T^{10} - 9820 p^{3} T^{11} + 1017 p^{4} T^{12} - 47 p^{5} T^{13} + 47 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 13 T + 88 T^{2} - 659 T^{3} + 6712 T^{4} - 88400 T^{5} + 693262 T^{6} - 3514828 T^{7} + 26130695 T^{8} - 3514828 p T^{9} + 693262 p^{2} T^{10} - 88400 p^{3} T^{11} + 6712 p^{4} T^{12} - 659 p^{5} T^{13} + 88 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - T + 168 T^{2} - 89 T^{3} + 14153 T^{4} - 89 p T^{5} + 168 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 11 T + 83 T^{2} + 331 T^{3} + 1907 T^{4} - 19610 T^{5} + 16782 T^{6} + 1589858 T^{7} + 30712905 T^{8} + 1589858 p T^{9} + 16782 p^{2} T^{10} - 19610 p^{3} T^{11} + 1907 p^{4} T^{12} + 331 p^{5} T^{13} + 83 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 7 T - 98 T^{2} - 469 T^{3} + 6176 T^{4} + 19502 T^{5} + 17990 T^{6} - 1098062 T^{7} - 36323381 T^{8} - 1098062 p T^{9} + 17990 p^{2} T^{10} + 19502 p^{3} T^{11} + 6176 p^{4} T^{12} - 469 p^{5} T^{13} - 98 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 22 T + 125 T^{2} - 10 T^{3} + 6225 T^{4} + 73966 T^{5} - 403353 T^{6} - 7990450 T^{7} - 58812140 T^{8} - 7990450 p T^{9} - 403353 p^{2} T^{10} + 73966 p^{3} T^{11} + 6225 p^{4} T^{12} - 10 p^{5} T^{13} + 125 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 21 T + 109 T^{2} + 615 T^{3} + 22511 T^{4} + 220998 T^{5} + 720980 T^{6} + 10036422 T^{7} + 165244057 T^{8} + 10036422 p T^{9} + 720980 p^{2} T^{10} + 220998 p^{3} T^{11} + 22511 p^{4} T^{12} + 615 p^{5} T^{13} + 109 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 10 T + 3 p T^{2} + 2360 T^{3} + 31893 T^{4} + 2360 p T^{5} + 3 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 31 T + 236 T^{2} + 2512 T^{3} - 50694 T^{4} + 179387 T^{5} + 1682790 T^{6} - 13200830 T^{7} + 28675747 T^{8} - 13200830 p T^{9} + 1682790 p^{2} T^{10} + 179387 p^{3} T^{11} - 50694 p^{4} T^{12} + 2512 p^{5} T^{13} + 236 p^{6} T^{14} - 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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
−5.83977773458547115445625501047, −5.63985621624884816023719803835, −5.63387844351742828677188915659, −5.58198562981746525203180816587, −5.21426843447885058762701864968, −4.87352855858696915598483103451, −4.84477144858693161554434443541, −4.82705091919757726414983975721, −4.65309934059947441333179003513, −4.60058522220350842973082740401, −4.23527958756584545564956852580, −4.15993027818074448142367528828, −3.97060061287208196767713201781, −3.69985425966542054458225769682, −3.65582911305122491411455168921, −3.54102272851467987681702733628, −2.90990595033121497270647484969, −2.85066806787227402372414792078, −2.74560675947307042768284215081, −2.65329692448641264645435422450, −2.27465340670819353207486826212, −2.07205358455758633746043484407, −1.72302966651076402732363805623, −1.17355736768305559697348692290, −1.05318655451302162407312282274, 1.05318655451302162407312282274, 1.17355736768305559697348692290, 1.72302966651076402732363805623, 2.07205358455758633746043484407, 2.27465340670819353207486826212, 2.65329692448641264645435422450, 2.74560675947307042768284215081, 2.85066806787227402372414792078, 2.90990595033121497270647484969, 3.54102272851467987681702733628, 3.65582911305122491411455168921, 3.69985425966542054458225769682, 3.97060061287208196767713201781, 4.15993027818074448142367528828, 4.23527958756584545564956852580, 4.60058522220350842973082740401, 4.65309934059947441333179003513, 4.82705091919757726414983975721, 4.84477144858693161554434443541, 4.87352855858696915598483103451, 5.21426843447885058762701864968, 5.58198562981746525203180816587, 5.63387844351742828677188915659, 5.63985621624884816023719803835, 5.83977773458547115445625501047
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.