Dirichlet series
| L(s) = 1 | + 3·2-s + 3-s + 4-s + 8·5-s + 3·6-s − 5·8-s − 2·9-s + 24·10-s − 8·11-s + 12-s + 14·13-s + 8·15-s − 6·16-s + 5·17-s − 6·18-s + 19-s + 8·20-s − 24·22-s − 2·23-s − 5·24-s + 36·25-s + 42·26-s − 5·27-s + 26·29-s + 24·30-s + 2·31-s − 3·32-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 1/2·4-s + 3.57·5-s + 1.22·6-s − 1.76·8-s − 2/3·9-s + 7.58·10-s − 2.41·11-s + 0.288·12-s + 3.88·13-s + 2.06·15-s − 3/2·16-s + 1.21·17-s − 1.41·18-s + 0.229·19-s + 1.78·20-s − 5.11·22-s − 0.417·23-s − 1.02·24-s + 36/5·25-s + 8.23·26-s − 0.962·27-s + 4.82·29-s + 4.38·30-s + 0.359·31-s − 0.530·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 7^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.59924\times 10^{10}\) |
| Root analytic conductor: | \(4.63893\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 7^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(257.9741895\) |
| \(L(\frac12)\) | \(\approx\) | \(257.9741895\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{8} \) |
| 7 | \( 1 \) | |
| 11 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} - p^{4} T^{3} + 31 T^{4} - 13 p^{2} T^{5} + 43 p T^{6} - 123 T^{7} + 183 T^{8} - 123 p T^{9} + 43 p^{3} T^{10} - 13 p^{5} T^{11} + 31 p^{4} T^{12} - p^{9} T^{13} + p^{9} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - T + p T^{2} + p T^{4} - 10 T^{5} + 13 T^{6} - 13 p T^{7} - 8 p T^{8} - 13 p^{2} T^{9} + 13 p^{2} T^{10} - 10 p^{3} T^{11} + p^{5} T^{12} + p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 14 T + 148 T^{2} - 1124 T^{3} + 7362 T^{4} - 40298 T^{5} + 196824 T^{6} - 837114 T^{7} + 3217139 T^{8} - 837114 p T^{9} + 196824 p^{2} T^{10} - 40298 p^{3} T^{11} + 7362 p^{4} T^{12} - 1124 p^{5} T^{13} + 148 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 5 T + 71 T^{2} - 314 T^{3} + 165 p T^{4} - 10872 T^{5} + 75305 T^{6} - 256905 T^{7} + 1480196 T^{8} - 256905 p T^{9} + 75305 p^{2} T^{10} - 10872 p^{3} T^{11} + 165 p^{5} T^{12} - 314 p^{5} T^{13} + 71 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - T + 94 T^{2} - 113 T^{3} + 4337 T^{4} - 5441 T^{5} + 133546 T^{6} - 154201 T^{7} + 2972300 T^{8} - 154201 p T^{9} + 133546 p^{2} T^{10} - 5441 p^{3} T^{11} + 4337 p^{4} T^{12} - 113 p^{5} T^{13} + 94 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 106 T^{2} + 41 T^{3} + 4917 T^{4} - 5545 T^{5} + 146986 T^{6} - 319128 T^{7} + 3579036 T^{8} - 319128 p T^{9} + 146986 p^{2} T^{10} - 5545 p^{3} T^{11} + 4917 p^{4} T^{12} + 41 p^{5} T^{13} + 106 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 26 T + 461 T^{2} - 5778 T^{3} + 59369 T^{4} - 503798 T^{5} + 3723699 T^{6} - 23921902 T^{7} + 137189276 T^{8} - 23921902 p T^{9} + 3723699 p^{2} T^{10} - 503798 p^{3} T^{11} + 59369 p^{4} T^{12} - 5778 p^{5} T^{13} + 461 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 2 T + 152 T^{2} - 157 T^{3} + 11342 T^{4} - 3607 T^{5} + 549004 T^{6} + 38893 T^{7} + 19496643 T^{8} + 38893 p T^{9} + 549004 p^{2} T^{10} - 3607 p^{3} T^{11} + 11342 p^{4} T^{12} - 157 p^{5} T^{13} + 152 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T + 46 T^{2} + 197 T^{3} + 2207 T^{4} + 12883 T^{5} + 109970 T^{6} + 520439 T^{7} + 3360960 T^{8} + 520439 p T^{9} + 109970 p^{2} T^{10} + 12883 p^{3} T^{11} + 2207 p^{4} T^{12} + 197 p^{5} T^{13} + 46 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 3 T + 173 T^{2} + 634 T^{3} + 15413 T^{4} + 53824 T^{5} + 947743 T^{6} + 2884075 T^{7} + 44008620 T^{8} + 2884075 p T^{9} + 947743 p^{2} T^{10} + 53824 p^{3} T^{11} + 15413 p^{4} T^{12} + 634 p^{5} T^{13} + 173 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 4 T + 207 T^{2} - 718 T^{3} + 20832 T^{4} - 59766 T^{5} + 1363586 T^{6} - 3306898 T^{7} + 66347145 T^{8} - 3306898 p T^{9} + 1363586 p^{2} T^{10} - 59766 p^{3} T^{11} + 20832 p^{4} T^{12} - 718 p^{5} T^{13} + 207 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - T + 133 T^{2} - 10 T^{3} + 10829 T^{4} + 6280 T^{5} + 638835 T^{6} + 15231 p T^{7} + 31170148 T^{8} + 15231 p^{2} T^{9} + 638835 p^{2} T^{10} + 6280 p^{3} T^{11} + 10829 p^{4} T^{12} - 10 p^{5} T^{13} + 133 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 26 T + 611 T^{2} - 9264 T^{3} + 127487 T^{4} - 1382604 T^{5} + 13864489 T^{6} - 116735106 T^{7} + 919539912 T^{8} - 116735106 p T^{9} + 13864489 p^{2} T^{10} - 1382604 p^{3} T^{11} + 127487 p^{4} T^{12} - 9264 p^{5} T^{13} + 611 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 19 T + 465 T^{2} + 5598 T^{3} + 81158 T^{4} + 745793 T^{5} + 8304976 T^{6} + 63224871 T^{7} + 582941581 T^{8} + 63224871 p T^{9} + 8304976 p^{2} T^{10} + 745793 p^{3} T^{11} + 81158 p^{4} T^{12} + 5598 p^{5} T^{13} + 465 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 308 T^{2} - 223 T^{3} + 46545 T^{4} - 55919 T^{5} + 4601696 T^{6} - 6052214 T^{7} + 327168036 T^{8} - 6052214 p T^{9} + 4601696 p^{2} T^{10} - 55919 p^{3} T^{11} + 46545 p^{4} T^{12} - 223 p^{5} T^{13} + 308 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 + 13 T + 398 T^{2} + 4313 T^{3} + 74671 T^{4} + 665357 T^{5} + 8595962 T^{6} + 64465721 T^{7} + 680524048 T^{8} + 64465721 p T^{9} + 8595962 p^{2} T^{10} + 665357 p^{3} T^{11} + 74671 p^{4} T^{12} + 4313 p^{5} T^{13} + 398 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 9 T + 301 T^{2} + 2750 T^{3} + 50086 T^{4} + 432311 T^{5} + 5835944 T^{6} + 43247021 T^{7} + 486890597 T^{8} + 43247021 p T^{9} + 5835944 p^{2} T^{10} + 432311 p^{3} T^{11} + 50086 p^{4} T^{12} + 2750 p^{5} T^{13} + 301 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 11 T + 388 T^{2} - 4289 T^{3} + 76746 T^{4} - 770936 T^{5} + 9828552 T^{6} - 84324936 T^{7} + 862766555 T^{8} - 84324936 p T^{9} + 9828552 p^{2} T^{10} - 770936 p^{3} T^{11} + 76746 p^{4} T^{12} - 4289 p^{5} T^{13} + 388 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 8 T + 286 T^{2} + 1797 T^{3} + 42445 T^{4} + 222987 T^{5} + 4576970 T^{6} + 21737790 T^{7} + 396616228 T^{8} + 21737790 p T^{9} + 4576970 p^{2} T^{10} + 222987 p^{3} T^{11} + 42445 p^{4} T^{12} + 1797 p^{5} T^{13} + 286 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 32 T + 894 T^{2} - 17215 T^{3} + 289362 T^{4} - 4015957 T^{5} + 49859860 T^{6} - 535436635 T^{7} + 5217653411 T^{8} - 535436635 p T^{9} + 49859860 p^{2} T^{10} - 4015957 p^{3} T^{11} + 289362 p^{4} T^{12} - 17215 p^{5} T^{13} + 894 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 5 T + 299 T^{2} - 3412 T^{3} + 50404 T^{4} - 688693 T^{5} + 7826676 T^{6} - 76130745 T^{7} + 887461699 T^{8} - 76130745 p T^{9} + 7826676 p^{2} T^{10} - 688693 p^{3} T^{11} + 50404 p^{4} T^{12} - 3412 p^{5} T^{13} + 299 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 9 T + 250 T^{2} - 1355 T^{3} + 18787 T^{4} - 159857 T^{5} + 1901202 T^{6} - 38051579 T^{7} + 295132128 T^{8} - 38051579 p T^{9} + 1901202 p^{2} T^{10} - 159857 p^{3} T^{11} + 18787 p^{4} T^{12} - 1355 p^{5} T^{13} + 250 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.59597028535846707553561585208, −3.48010228315342660022175508496, −3.38632194669633238821776146148, −3.28169740664187136048205841883, −3.26282351207271964538819880031, −3.07729411125138248724784972078, −2.97781162665601635202013076140, −2.94379129351713899393564888335, −2.80135659613892074754603733760, −2.74414367324800279081540188510, −2.49360489457940496476849823295, −2.21932806215075507141211782282, −2.21668960235290726405892461421, −2.17448694888949445200605090925, −2.04203097334280111109037083130, −1.84406875703634420617952453683, −1.65790176653506851719202773251, −1.56784431073552010984795923440, −1.51958758378998867683686312955, −1.05267491498905046360166366469, −1.04610837227610474902528868308, −0.76984389689048114866546437130, −0.68452516381464898816598356508, −0.60590353295889219895515098998, −0.54778690774648245258095941197, 0.54778690774648245258095941197, 0.60590353295889219895515098998, 0.68452516381464898816598356508, 0.76984389689048114866546437130, 1.04610837227610474902528868308, 1.05267491498905046360166366469, 1.51958758378998867683686312955, 1.56784431073552010984795923440, 1.65790176653506851719202773251, 1.84406875703634420617952453683, 2.04203097334280111109037083130, 2.17448694888949445200605090925, 2.21668960235290726405892461421, 2.21932806215075507141211782282, 2.49360489457940496476849823295, 2.74414367324800279081540188510, 2.80135659613892074754603733760, 2.94379129351713899393564888335, 2.97781162665601635202013076140, 3.07729411125138248724784972078, 3.26282351207271964538819880031, 3.28169740664187136048205841883, 3.38632194669633238821776146148, 3.48010228315342660022175508496, 3.59597028535846707553561585208
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.