Dirichlet series
| L(s) = 1 | − 2·3-s + 2·5-s + 8·7-s + 9-s + 2·13-s − 4·15-s − 10·17-s − 6·19-s − 16·21-s − 8·23-s + 5·25-s − 20·29-s + 10·31-s + 16·35-s + 26·37-s − 4·39-s + 6·41-s + 12·43-s + 2·45-s − 12·47-s + 49·49-s + 20·51-s + 14·53-s + 12·57-s + 12·59-s − 14·61-s + 8·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 3.02·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s − 2.42·17-s − 1.37·19-s − 3.49·21-s − 1.66·23-s + 25-s − 3.71·29-s + 1.79·31-s + 2.70·35-s + 4.27·37-s − 0.640·39-s + 0.937·41-s + 1.82·43-s + 0.298·45-s − 1.75·47-s + 7·49-s + 2.80·51-s + 1.92·53-s + 1.58·57-s + 1.56·59-s − 1.79·61-s + 1.00·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{40} \cdot 3^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.55580\times 10^{7}\) |
| Root analytic conductor: | \(2.90382\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{40} \cdot 3^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.799722894\) |
| \(L(\frac12)\) | \(\approx\) | \(7.799722894\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 + 31 T^{2} + 41 p T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 2 T - T^{2} + 2 T^{3} + 11 T^{4} - 22 p T^{5} + 37 p T^{6} - 2 p^{2} T^{7} + 8 p^{2} T^{8} - 2 p^{3} T^{9} + 37 p^{3} T^{10} - 22 p^{4} T^{11} + 11 p^{4} T^{12} + 2 p^{5} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 8 T + 15 T^{2} + 8 p T^{3} - 254 T^{4} - 52 T^{5} + 1571 T^{6} + 250 T^{7} - 11581 T^{8} + 250 p T^{9} + 1571 p^{2} T^{10} - 52 p^{3} T^{11} - 254 p^{4} T^{12} + 8 p^{6} T^{13} + 15 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 25 p T^{5} - 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 10 T + 10 T^{2} - 270 T^{3} - 1289 T^{4} - 350 T^{5} + 12280 T^{6} + 34840 T^{7} + 63921 T^{8} + 34840 p T^{9} + 12280 p^{2} T^{10} - 350 p^{3} T^{11} - 1289 p^{4} T^{12} - 270 p^{5} T^{13} + 10 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 6 T - 12 T^{2} - 86 T^{3} + 573 T^{4} + 1206 T^{5} - 18160 T^{6} - 26260 T^{7} + 242161 T^{8} - 26260 p T^{9} - 18160 p^{2} T^{10} + 1206 p^{3} T^{11} + 573 p^{4} T^{12} - 86 p^{5} T^{13} - 12 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 4 T + 80 T^{2} + 224 T^{3} + 2581 T^{4} + 224 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 20 T + 166 T^{2} + 860 T^{3} + 4755 T^{4} + 33780 T^{5} + 197064 T^{6} + 761200 T^{7} + 2853189 T^{8} + 761200 p T^{9} + 197064 p^{2} T^{10} + 33780 p^{3} T^{11} + 4755 p^{4} T^{12} + 860 p^{5} T^{13} + 166 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 10 T + 4 T^{2} + 330 T^{3} - 1755 T^{4} + 710 T^{5} + 38296 T^{6} - 49900 T^{7} - 975191 T^{8} - 49900 p T^{9} + 38296 p^{2} T^{10} + 710 p^{3} T^{11} - 1755 p^{4} T^{12} + 330 p^{5} T^{13} + 4 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 26 T + 269 T^{2} - 1294 T^{3} + 1484 T^{4} + 21636 T^{5} - 257957 T^{6} + 2267268 T^{7} - 15721817 T^{8} + 2267268 p T^{9} - 257957 p^{2} T^{10} + 21636 p^{3} T^{11} + 1484 p^{4} T^{12} - 1294 p^{5} T^{13} + 269 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 6 T - 31 T^{2} + 242 T^{3} - 160 T^{4} - 7076 T^{5} + 71495 T^{6} + 240030 T^{7} - 5091109 T^{8} + 240030 p T^{9} + 71495 p^{2} T^{10} - 7076 p^{3} T^{11} - 160 p^{4} T^{12} + 242 p^{5} T^{13} - 31 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 6 T + 100 T^{2} - 716 T^{3} + 4901 T^{4} - 716 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 12 T + 99 T^{2} + 932 T^{3} + 7773 T^{4} + 57292 T^{5} + 474221 T^{6} + 3986676 T^{7} + 27952628 T^{8} + 3986676 p T^{9} + 474221 p^{2} T^{10} + 57292 p^{3} T^{11} + 7773 p^{4} T^{12} + 932 p^{5} T^{13} + 99 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 14 T + 90 T^{2} - 718 T^{3} + 6511 T^{4} - 76886 T^{5} + 691704 T^{6} - 4321800 T^{7} + 29508369 T^{8} - 4321800 p T^{9} + 691704 p^{2} T^{10} - 76886 p^{3} T^{11} + 6511 p^{4} T^{12} - 718 p^{5} T^{13} + 90 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 12 T + 16 T^{2} + 4 p T^{3} + 1505 T^{4} - 15612 T^{5} - 121620 T^{6} + 2376120 T^{7} - 19015459 T^{8} + 2376120 p T^{9} - 121620 p^{2} T^{10} - 15612 p^{3} T^{11} + 1505 p^{4} T^{12} + 4 p^{6} T^{13} + 16 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T - 21 T^{2} - 1008 T^{3} - 3545 T^{4} + 26684 T^{5} + 497385 T^{6} + 343070 T^{7} - 30611544 T^{8} + 343070 p T^{9} + 497385 p^{2} T^{10} + 26684 p^{3} T^{11} - 3545 p^{4} T^{12} - 1008 p^{5} T^{13} - 21 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 6 T + 219 T^{2} + 882 T^{3} + 19959 T^{4} + 882 p T^{5} + 219 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 24 T + 149 T^{2} + 128 T^{3} + 3365 T^{4} - 33344 T^{5} - 701605 T^{6} - 4200 p T^{7} + 87081956 T^{8} - 4200 p^{2} T^{9} - 701605 p^{2} T^{10} - 33344 p^{3} T^{11} + 3365 p^{4} T^{12} + 128 p^{5} T^{13} + 149 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 32 T + 418 T^{2} + 3580 T^{3} + 33755 T^{4} + 272368 T^{5} + 1116824 T^{6} + 4568960 T^{7} + 52288789 T^{8} + 4568960 p T^{9} + 1116824 p^{2} T^{10} + 272368 p^{3} T^{11} + 33755 p^{4} T^{12} + 3580 p^{5} T^{13} + 418 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 2 T - 71 T^{2} - 434 T^{3} + 6062 T^{4} + 51932 T^{5} - 204547 T^{6} - 781034 T^{7} + 32478331 T^{8} - 781034 p T^{9} - 204547 p^{2} T^{10} + 51932 p^{3} T^{11} + 6062 p^{4} T^{12} - 434 p^{5} T^{13} - 71 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 12 T + 17 T^{2} - 1036 T^{3} + 20066 T^{4} - 122792 T^{5} + 1112705 T^{6} - 15564648 T^{7} + 148848239 T^{8} - 15564648 p T^{9} + 1112705 p^{2} T^{10} - 122792 p^{3} T^{11} + 20066 p^{4} T^{12} - 1036 p^{5} T^{13} + 17 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 18 T + 262 T^{2} + 1656 T^{3} + 16839 T^{4} + 1656 p T^{5} + 262 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 30 T + 215 T^{2} + 2140 T^{3} - 27569 T^{4} - 285160 T^{5} + 5186565 T^{6} + 16773930 T^{7} - 699217624 T^{8} + 16773930 p T^{9} + 5186565 p^{2} T^{10} - 285160 p^{3} T^{11} - 27569 p^{4} T^{12} + 2140 p^{5} T^{13} + 215 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.35392248827332599997513591852, −4.26822628316406260351364750749, −4.03710396825777587494687581863, −3.92696197197316548701225875632, −3.90003137528848642471622211449, −3.84990212037490553888075563635, −3.53456428675265643804642106895, −3.20555260902212454890285801749, −3.14111110976705045010063148626, −2.99449888683246874132118539611, −2.76934255034691068012547728104, −2.53235545121809821908856887679, −2.50922601652731831814350415026, −2.47403146385608313338653953951, −2.16585985652264021146585026708, −1.96571569627372972700757676444, −1.93817366150440658072264726770, −1.80512844153641798411690453084, −1.79386222813391335310605784275, −1.55597260703134258136972974784, −1.20062957936781925095160120098, −0.904813459292764989817490559228, −0.75835657283746663702654077482, −0.61157619547160499794272630144, −0.35446452375689316049136812544, 0.35446452375689316049136812544, 0.61157619547160499794272630144, 0.75835657283746663702654077482, 0.904813459292764989817490559228, 1.20062957936781925095160120098, 1.55597260703134258136972974784, 1.79386222813391335310605784275, 1.80512844153641798411690453084, 1.93817366150440658072264726770, 1.96571569627372972700757676444, 2.16585985652264021146585026708, 2.47403146385608313338653953951, 2.50922601652731831814350415026, 2.53235545121809821908856887679, 2.76934255034691068012547728104, 2.99449888683246874132118539611, 3.14111110976705045010063148626, 3.20555260902212454890285801749, 3.53456428675265643804642106895, 3.84990212037490553888075563635, 3.90003137528848642471622211449, 3.92696197197316548701225875632, 4.03710396825777587494687581863, 4.26822628316406260351364750749, 4.35392248827332599997513591852
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.