Dirichlet series
| L(s) = 1 | − 5·4-s − 8·5-s + 4·7-s + 2·8-s + 6·11-s − 8·13-s + 7·16-s − 2·17-s − 10·19-s + 40·20-s + 30·23-s + 36·25-s − 20·28-s + 8·29-s − 10·31-s − 14·32-s − 32·35-s − 8·37-s − 16·40-s + 6·41-s − 30·44-s + 18·47-s − 7·49-s + 40·52-s + 14·53-s − 48·55-s + 8·56-s + ⋯ |
| L(s) = 1 | − 5/2·4-s − 3.57·5-s + 1.51·7-s + 0.707·8-s + 1.80·11-s − 2.21·13-s + 7/4·16-s − 0.485·17-s − 2.29·19-s + 8.94·20-s + 6.25·23-s + 36/5·25-s − 3.77·28-s + 1.48·29-s − 1.79·31-s − 2.47·32-s − 5.40·35-s − 1.31·37-s − 2.52·40-s + 0.937·41-s − 4.52·44-s + 2.62·47-s − 49-s + 5.54·52-s + 1.92·53-s − 6.47·55-s + 1.06·56-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.75896\times 10^{12}\) |
| Root analytic conductor: | \(6.48392\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{32} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.557367585\) |
| \(L(\frac12)\) | \(\approx\) | \(6.557367585\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{8} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 + 5 T^{2} - p T^{3} + 9 p T^{4} - 3 p T^{5} + 53 T^{6} - 9 p T^{7} + 57 p T^{8} - 9 p^{2} T^{9} + 53 p^{2} T^{10} - 3 p^{4} T^{11} + 9 p^{5} T^{12} - p^{6} T^{13} + 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 T + 23 T^{2} - 74 T^{3} + 216 T^{4} - 454 T^{5} + 794 T^{6} - 690 T^{7} + 1186 T^{8} - 690 p T^{9} + 794 p^{2} T^{10} - 454 p^{3} T^{11} + 216 p^{4} T^{12} - 74 p^{5} T^{13} + 23 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 6 T + 7 p T^{2} - 362 T^{3} + 2613 T^{4} - 9972 T^{5} + 52595 T^{6} - 165936 T^{7} + 700437 T^{8} - 165936 p T^{9} + 52595 p^{2} T^{10} - 9972 p^{3} T^{11} + 2613 p^{4} T^{12} - 362 p^{5} T^{13} + 7 p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T + 73 T^{2} + 84 T^{3} + 2586 T^{4} + 894 T^{5} + 60360 T^{6} - 10838 T^{7} + 1113666 T^{8} - 10838 p T^{9} + 60360 p^{2} T^{10} + 894 p^{3} T^{11} + 2586 p^{4} T^{12} + 84 p^{5} T^{13} + 73 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 10 T + 94 T^{2} + 618 T^{3} + 3797 T^{4} + 20188 T^{5} + 101967 T^{6} + 24766 p T^{7} + 2110979 T^{8} + 24766 p^{2} T^{9} + 101967 p^{2} T^{10} + 20188 p^{3} T^{11} + 3797 p^{4} T^{12} + 618 p^{5} T^{13} + 94 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 30 T + 519 T^{2} - 6426 T^{3} + 62897 T^{4} - 508500 T^{5} + 3491022 T^{6} - 20646204 T^{7} + 106111866 T^{8} - 20646204 p T^{9} + 3491022 p^{2} T^{10} - 508500 p^{3} T^{11} + 62897 p^{4} T^{12} - 6426 p^{5} T^{13} + 519 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 8 T + 150 T^{2} - 748 T^{3} + 9111 T^{4} - 34488 T^{5} + 386857 T^{6} - 1321060 T^{7} + 13057491 T^{8} - 1321060 p T^{9} + 386857 p^{2} T^{10} - 34488 p^{3} T^{11} + 9111 p^{4} T^{12} - 748 p^{5} T^{13} + 150 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 10 T + 149 T^{2} + 1142 T^{3} + 11678 T^{4} + 74726 T^{5} + 593067 T^{6} + 3263610 T^{7} + 21777922 T^{8} + 3263610 p T^{9} + 593067 p^{2} T^{10} + 74726 p^{3} T^{11} + 11678 p^{4} T^{12} + 1142 p^{5} T^{13} + 149 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 8 T + 266 T^{2} + 1714 T^{3} + 31619 T^{4} + 168652 T^{5} + 2219457 T^{6} + 9805620 T^{7} + 100830196 T^{8} + 9805620 p T^{9} + 2219457 p^{2} T^{10} + 168652 p^{3} T^{11} + 31619 p^{4} T^{12} + 1714 p^{5} T^{13} + 266 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 6 T + 183 T^{2} - 1374 T^{3} + 17147 T^{4} - 137538 T^{5} + 1106157 T^{6} - 8308968 T^{7} + 52651425 T^{8} - 8308968 p T^{9} + 1106157 p^{2} T^{10} - 137538 p^{3} T^{11} + 17147 p^{4} T^{12} - 1374 p^{5} T^{13} + 183 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 243 T^{2} + 434 T^{3} + 26436 T^{4} + 85658 T^{5} + 1791058 T^{6} + 7026774 T^{7} + 88074226 T^{8} + 7026774 p T^{9} + 1791058 p^{2} T^{10} + 85658 p^{3} T^{11} + 26436 p^{4} T^{12} + 434 p^{5} T^{13} + 243 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 18 T + 362 T^{2} - 4456 T^{3} + 55389 T^{4} - 527502 T^{5} + 4930241 T^{6} - 37953714 T^{7} + 284360484 T^{8} - 37953714 p T^{9} + 4930241 p^{2} T^{10} - 527502 p^{3} T^{11} + 55389 p^{4} T^{12} - 4456 p^{5} T^{13} + 362 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 14 T + 373 T^{2} - 4284 T^{3} + 64094 T^{4} - 600602 T^{5} + 6509124 T^{6} - 49887058 T^{7} + 425223178 T^{8} - 49887058 p T^{9} + 6509124 p^{2} T^{10} - 600602 p^{3} T^{11} + 64094 p^{4} T^{12} - 4284 p^{5} T^{13} + 373 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 30 T + 643 T^{2} - 9762 T^{3} + 123511 T^{4} - 1317150 T^{5} + 12548341 T^{6} - 108077268 T^{7} + 862875721 T^{8} - 108077268 p T^{9} + 12548341 p^{2} T^{10} - 1317150 p^{3} T^{11} + 123511 p^{4} T^{12} - 9762 p^{5} T^{13} + 643 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 18 T + 469 T^{2} + 5940 T^{3} + 91720 T^{4} + 908316 T^{5} + 10421368 T^{6} + 84115986 T^{7} + 773946376 T^{8} + 84115986 p T^{9} + 10421368 p^{2} T^{10} + 908316 p^{3} T^{11} + 91720 p^{4} T^{12} + 5940 p^{5} T^{13} + 469 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 6 T + 499 T^{2} + 2628 T^{3} + 111136 T^{4} + 504342 T^{5} + 14436802 T^{6} + 55019064 T^{7} + 1196611660 T^{8} + 55019064 p T^{9} + 14436802 p^{2} T^{10} + 504342 p^{3} T^{11} + 111136 p^{4} T^{12} + 2628 p^{5} T^{13} + 499 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 16 T + 192 T^{2} - 1594 T^{3} + 20483 T^{4} - 163034 T^{5} + 1620237 T^{6} - 14042346 T^{7} + 157220931 T^{8} - 14042346 p T^{9} + 1620237 p^{2} T^{10} - 163034 p^{3} T^{11} + 20483 p^{4} T^{12} - 1594 p^{5} T^{13} + 192 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 12 T + 335 T^{2} - 2694 T^{3} + 48556 T^{4} - 320670 T^{5} + 5133458 T^{6} - 31786890 T^{7} + 435575470 T^{8} - 31786890 p T^{9} + 5133458 p^{2} T^{10} - 320670 p^{3} T^{11} + 48556 p^{4} T^{12} - 2694 p^{5} T^{13} + 335 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 30 T + 749 T^{2} + 13692 T^{3} + 218308 T^{4} + 2905056 T^{5} + 34851080 T^{6} + 363577830 T^{7} + 3443672992 T^{8} + 363577830 p T^{9} + 34851080 p^{2} T^{10} + 2905056 p^{3} T^{11} + 218308 p^{4} T^{12} + 13692 p^{5} T^{13} + 749 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 26 T + 681 T^{2} - 11656 T^{3} + 188250 T^{4} - 2474634 T^{5} + 30071380 T^{6} - 316801234 T^{7} + 3073283430 T^{8} - 316801234 p T^{9} + 30071380 p^{2} T^{10} - 2474634 p^{3} T^{11} + 188250 p^{4} T^{12} - 11656 p^{5} T^{13} + 681 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 14 T + 412 T^{2} - 4926 T^{3} + 81797 T^{4} - 925262 T^{5} + 11120919 T^{6} - 1326908 p T^{7} + 1139762455 T^{8} - 1326908 p^{2} T^{9} + 11120919 p^{2} T^{10} - 925262 p^{3} T^{11} + 81797 p^{4} T^{12} - 4926 p^{5} T^{13} + 412 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 44 T + 1169 T^{2} - 23656 T^{3} + 401682 T^{4} - 5924198 T^{5} + 77040656 T^{6} - 890031960 T^{7} + 9236126554 T^{8} - 890031960 p T^{9} + 77040656 p^{2} T^{10} - 5924198 p^{3} T^{11} + 401682 p^{4} T^{12} - 23656 p^{5} T^{13} + 1169 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.54113354153661608481513136424, −3.15631017289374982382286189043, −3.15485052456679175959658491086, −3.10270654359462162666575256545, −2.89563963240287666383049251866, −2.81212795003026339769509343765, −2.72394341355458053157943504514, −2.66057577378976024557911417406, −2.50238411264204925444706461140, −2.36540950834451729317132576793, −2.30109524891364726611743029474, −2.07870420850571787883878883825, −1.77625389614338266294411455242, −1.73670878181021201593852428510, −1.69857668308643484450810244831, −1.55424893511907565489851750238, −1.45427701058472426562690685057, −1.13555368086993064797860364063, −1.00817523090904100769633596941, −0.803643829521840403699657353839, −0.67154601238871733358987473525, −0.55275320465706242443458118147, −0.43793449324495058005503781656, −0.43016528781253750637890629920, −0.37271078690748911518311340033, 0.37271078690748911518311340033, 0.43016528781253750637890629920, 0.43793449324495058005503781656, 0.55275320465706242443458118147, 0.67154601238871733358987473525, 0.803643829521840403699657353839, 1.00817523090904100769633596941, 1.13555368086993064797860364063, 1.45427701058472426562690685057, 1.55424893511907565489851750238, 1.69857668308643484450810244831, 1.73670878181021201593852428510, 1.77625389614338266294411455242, 2.07870420850571787883878883825, 2.30109524891364726611743029474, 2.36540950834451729317132576793, 2.50238411264204925444706461140, 2.66057577378976024557911417406, 2.72394341355458053157943504514, 2.81212795003026339769509343765, 2.89563963240287666383049251866, 3.10270654359462162666575256545, 3.15485052456679175959658491086, 3.15631017289374982382286189043, 3.54113354153661608481513136424
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.