Dirichlet series
| L(s) = 1 | − 8·2-s + 3-s + 36·4-s + 8·5-s − 8·6-s + 11·7-s − 120·8-s − 7·9-s − 64·10-s + 36·12-s − 8·13-s − 88·14-s + 8·15-s + 330·16-s + 7·17-s + 56·18-s − 2·19-s + 288·20-s + 11·21-s + 8·23-s − 120·24-s + 36·25-s + 64·26-s − 6·27-s + 396·28-s − 29-s − 64·30-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 0.577·3-s + 18·4-s + 3.57·5-s − 3.26·6-s + 4.15·7-s − 42.4·8-s − 7/3·9-s − 20.2·10-s + 10.3·12-s − 2.21·13-s − 23.5·14-s + 2.06·15-s + 82.5·16-s + 1.69·17-s + 13.1·18-s − 0.458·19-s + 64.3·20-s + 2.40·21-s + 1.66·23-s − 24.4·24-s + 36/5·25-s + 12.5·26-s − 1.15·27-s + 74.8·28-s − 0.185·29-s − 11.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14989\times 10^{12}\) |
| Root analytic conductor: | \(5.67271\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 13^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(12.55669910\) |
| \(L(\frac12)\) | \(\approx\) | \(12.55669910\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 5 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| 31 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 - T + 8 T^{2} - p^{2} T^{3} + 32 T^{4} - 13 p T^{5} + 11 p^{2} T^{6} - 101 T^{7} + 284 T^{8} - 101 p T^{9} + 11 p^{4} T^{10} - 13 p^{4} T^{11} + 32 p^{4} T^{12} - p^{7} T^{13} + 8 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 11 T + 83 T^{2} - 461 T^{3} + 2120 T^{4} - 8354 T^{5} + 29115 T^{6} - 90354 T^{7} + 252633 T^{8} - 90354 p T^{9} + 29115 p^{2} T^{10} - 8354 p^{3} T^{11} + 2120 p^{4} T^{12} - 461 p^{5} T^{13} + 83 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 45 T^{2} - 6 T^{3} + 1022 T^{4} - 134 T^{5} + 1462 p T^{6} - 1390 T^{7} + 197328 T^{8} - 1390 p T^{9} + 1462 p^{3} T^{10} - 134 p^{3} T^{11} + 1022 p^{4} T^{12} - 6 p^{5} T^{13} + 45 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 7 T + 114 T^{2} - 618 T^{3} + 5772 T^{4} - 25332 T^{5} + 174771 T^{6} - 637573 T^{7} + 3560205 T^{8} - 637573 p T^{9} + 174771 p^{2} T^{10} - 25332 p^{3} T^{11} + 5772 p^{4} T^{12} - 618 p^{5} T^{13} + 114 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 2 T + 85 T^{2} + 264 T^{3} + 3864 T^{4} + 13047 T^{5} + 6247 p T^{6} + 386184 T^{7} + 2611287 T^{8} + 386184 p T^{9} + 6247 p^{3} T^{10} + 13047 p^{3} T^{11} + 3864 p^{4} T^{12} + 264 p^{5} T^{13} + 85 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T + 78 T^{2} - 19 p T^{3} + 2973 T^{4} - 13665 T^{5} + 83315 T^{6} - 359654 T^{7} + 2101135 T^{8} - 359654 p T^{9} + 83315 p^{2} T^{10} - 13665 p^{3} T^{11} + 2973 p^{4} T^{12} - 19 p^{6} T^{13} + 78 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + T + 138 T^{2} + 132 T^{3} + 9086 T^{4} + 7770 T^{5} + 394011 T^{6} + 287431 T^{7} + 12885991 T^{8} + 287431 p T^{9} + 394011 p^{2} T^{10} + 7770 p^{3} T^{11} + 9086 p^{4} T^{12} + 132 p^{5} T^{13} + 138 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T + 114 T^{2} - 72 T^{3} + 8149 T^{4} - 5372 T^{5} + 437773 T^{6} - 277038 T^{7} + 18288227 T^{8} - 277038 p T^{9} + 437773 p^{2} T^{10} - 5372 p^{3} T^{11} + 8149 p^{4} T^{12} - 72 p^{5} T^{13} + 114 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 16 T + 251 T^{2} - 2037 T^{3} + 17036 T^{4} - 73858 T^{5} + 406456 T^{6} - 240459 T^{7} + 5258680 T^{8} - 240459 p T^{9} + 406456 p^{2} T^{10} - 73858 p^{3} T^{11} + 17036 p^{4} T^{12} - 2037 p^{5} T^{13} + 251 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 3 T + 87 T^{2} - 871 T^{3} + 6691 T^{4} - 55985 T^{5} + 458586 T^{6} - 3388311 T^{7} + 18778858 T^{8} - 3388311 p T^{9} + 458586 p^{2} T^{10} - 55985 p^{3} T^{11} + 6691 p^{4} T^{12} - 871 p^{5} T^{13} + 87 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 29 T + 515 T^{2} - 6567 T^{3} + 69086 T^{4} - 641148 T^{5} + 5483787 T^{6} - 43414700 T^{7} + 312865551 T^{8} - 43414700 p T^{9} + 5483787 p^{2} T^{10} - 641148 p^{3} T^{11} + 69086 p^{4} T^{12} - 6567 p^{5} T^{13} + 515 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 22 T + 539 T^{2} - 7751 T^{3} + 111104 T^{4} - 1186088 T^{5} + 12336020 T^{6} - 102672835 T^{7} + 828084456 T^{8} - 102672835 p T^{9} + 12336020 p^{2} T^{10} - 1186088 p^{3} T^{11} + 111104 p^{4} T^{12} - 7751 p^{5} T^{13} + 539 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 8 T + 280 T^{2} + 2031 T^{3} + 38093 T^{4} + 252715 T^{5} + 3346311 T^{6} + 20474660 T^{7} + 221488277 T^{8} + 20474660 p T^{9} + 3346311 p^{2} T^{10} + 252715 p^{3} T^{11} + 38093 p^{4} T^{12} + 2031 p^{5} T^{13} + 280 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 4 T + 169 T^{2} - 564 T^{3} + 12585 T^{4} - 4107 T^{5} + 363455 T^{6} + 3686371 T^{7} + 5278461 T^{8} + 3686371 p T^{9} + 363455 p^{2} T^{10} - 4107 p^{3} T^{11} + 12585 p^{4} T^{12} - 564 p^{5} T^{13} + 169 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 28 T + 680 T^{2} - 11156 T^{3} + 165752 T^{4} - 1986969 T^{5} + 22014833 T^{6} - 207831337 T^{7} + 1830370504 T^{8} - 207831337 p T^{9} + 22014833 p^{2} T^{10} - 1986969 p^{3} T^{11} + 165752 p^{4} T^{12} - 11156 p^{5} T^{13} + 680 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 4 T + 297 T^{2} - 1676 T^{3} + 42061 T^{4} - 337176 T^{5} + 3929630 T^{6} - 39076936 T^{7} + 295920522 T^{8} - 39076936 p T^{9} + 3929630 p^{2} T^{10} - 337176 p^{3} T^{11} + 42061 p^{4} T^{12} - 1676 p^{5} T^{13} + 297 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 39 T + 1003 T^{2} - 18777 T^{3} + 289296 T^{4} - 3778440 T^{5} + 43204278 T^{6} - 437307874 T^{7} + 3951024500 T^{8} - 437307874 p T^{9} + 43204278 p^{2} T^{10} - 3778440 p^{3} T^{11} + 289296 p^{4} T^{12} - 18777 p^{5} T^{13} + 1003 p^{6} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 16 T + 459 T^{2} + 4382 T^{3} + 62490 T^{4} + 244870 T^{5} + 1751978 T^{6} - 23487992 T^{7} - 128425572 T^{8} - 23487992 p T^{9} + 1751978 p^{2} T^{10} + 244870 p^{3} T^{11} + 62490 p^{4} T^{12} + 4382 p^{5} T^{13} + 459 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 25 T + 715 T^{2} - 10997 T^{3} + 184864 T^{4} - 25616 p T^{5} + 27052661 T^{6} - 254244608 T^{7} + 2669963989 T^{8} - 254244608 p T^{9} + 27052661 p^{2} T^{10} - 25616 p^{4} T^{11} + 184864 p^{4} T^{12} - 10997 p^{5} T^{13} + 715 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 21 T + 682 T^{2} - 8688 T^{3} + 154618 T^{4} - 1253676 T^{5} + 16849607 T^{6} - 92928843 T^{7} + 1385000209 T^{8} - 92928843 p T^{9} + 16849607 p^{2} T^{10} - 1253676 p^{3} T^{11} + 154618 p^{4} T^{12} - 8688 p^{5} T^{13} + 682 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 28 T + 544 T^{2} - 7186 T^{3} + 100594 T^{4} - 1263014 T^{5} + 15807736 T^{6} - 162323603 T^{7} + 1684704305 T^{8} - 162323603 p T^{9} + 15807736 p^{2} T^{10} - 1263014 p^{3} T^{11} + 100594 p^{4} T^{12} - 7186 p^{5} T^{13} + 544 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.50074517752737997523709672055, −2.86374915443084702322934526125, −2.78635615639217929898817786348, −2.78102380731806339753881722152, −2.77057358899172703538975936844, −2.74062999978843600952842795668, −2.69696589149387565795896157311, −2.45827894044374215814581520253, −2.44851908760400592195014992417, −2.13189950149679534495493981981, −2.11103963521513785413100578763, −2.01137546753659732172985641463, −1.98199277552403256729910949747, −1.85844225235591564901680990998, −1.81065582928333789140713531108, −1.68453761440257314613689645465, −1.54876181958223814183619633904, −1.25164262507789565640197973610, −1.09627964649368092505987505694, −0.947118898899582703595084304990, −0.846996275616194137084628863858, −0.811325791947373974236897668341, −0.61472584774665224096801135227, −0.51241314582257127417141450977, −0.42655359212020408579665098794, 0.42655359212020408579665098794, 0.51241314582257127417141450977, 0.61472584774665224096801135227, 0.811325791947373974236897668341, 0.846996275616194137084628863858, 0.947118898899582703595084304990, 1.09627964649368092505987505694, 1.25164262507789565640197973610, 1.54876181958223814183619633904, 1.68453761440257314613689645465, 1.81065582928333789140713531108, 1.85844225235591564901680990998, 1.98199277552403256729910949747, 2.01137546753659732172985641463, 2.11103963521513785413100578763, 2.13189950149679534495493981981, 2.44851908760400592195014992417, 2.45827894044374215814581520253, 2.69696589149387565795896157311, 2.74062999978843600952842795668, 2.77057358899172703538975936844, 2.78102380731806339753881722152, 2.78635615639217929898817786348, 2.86374915443084702322934526125, 3.50074517752737997523709672055
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.