Dirichlet series
| L(s) = 1 | + 4·3-s − 2·5-s + 8·7-s + 14·9-s − 4·11-s − 2·13-s − 8·15-s − 2·17-s + 2·19-s + 32·21-s − 2·23-s + 11·25-s + 40·27-s + 10·29-s − 24·31-s − 16·33-s − 16·35-s + 8·37-s − 8·39-s + 8·41-s − 18·43-s − 28·45-s + 6·47-s + 20·49-s − 8·51-s + 10·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 3.02·7-s + 14/3·9-s − 1.20·11-s − 0.554·13-s − 2.06·15-s − 0.485·17-s + 0.458·19-s + 6.98·21-s − 0.417·23-s + 11/5·25-s + 7.69·27-s + 1.85·29-s − 4.31·31-s − 2.78·33-s − 2.70·35-s + 1.31·37-s − 1.28·39-s + 1.24·41-s − 2.74·43-s − 4.17·45-s + 0.875·47-s + 20/7·49-s − 1.12·51-s + 1.37·53-s + 1.07·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.90106\times 10^{7}\) |
| Root analytic conductor: | \(3.11605\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1584931319\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1584931319\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 34 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( ( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 + 2 T - 7 T^{2} - 14 T^{3} + 11 T^{4} - 8 T^{5} - 24 p T^{6} + 152 T^{7} + 1206 T^{8} + 152 p T^{9} - 24 p^{3} T^{10} - 8 p^{3} T^{11} + 11 p^{4} T^{12} - 14 p^{5} T^{13} - 7 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( ( 1 - 4 T + 2 p T^{2} - 52 T^{3} + 162 T^{4} - 52 p T^{5} + 2 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 + 2 T + 35 T^{2} + 58 T^{3} + 544 T^{4} + 58 p T^{5} + 35 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 + 2 T - p T^{2} + 126 T^{3} + 369 T^{4} - 1428 T^{5} + 8854 T^{6} + 29920 T^{7} - 87190 T^{8} + 29920 p T^{9} + 8854 p^{2} T^{10} - 1428 p^{3} T^{11} + 369 p^{4} T^{12} + 126 p^{5} T^{13} - p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T - 25 T^{2} + 190 T^{3} + 749 T^{4} - 3932 T^{5} + 16866 T^{6} + 76880 T^{7} - 311142 T^{8} + 76880 p T^{9} + 16866 p^{2} T^{10} - 3932 p^{3} T^{11} + 749 p^{4} T^{12} + 190 p^{5} T^{13} - 25 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T - 29 T^{2} + 294 T^{3} + 1063 T^{4} - 7400 T^{5} + 44772 T^{6} + 203344 T^{7} - 992978 T^{8} + 203344 p T^{9} + 44772 p^{2} T^{10} - 7400 p^{3} T^{11} + 1063 p^{4} T^{12} + 294 p^{5} T^{13} - 29 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 10 T + 49 T^{2} - 250 T^{3} + 1083 T^{4} - 6040 T^{5} + 53864 T^{6} - 316600 T^{7} + 1540502 T^{8} - 316600 p T^{9} + 53864 p^{2} T^{10} - 6040 p^{3} T^{11} + 1083 p^{4} T^{12} - 250 p^{5} T^{13} + 49 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 12 T + 110 T^{2} + 788 T^{3} + 4506 T^{4} + 788 p T^{5} + 110 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 4 T + 110 T^{2} - 380 T^{3} + 5386 T^{4} - 380 p T^{5} + 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 8 T - 2 p T^{2} + 560 T^{3} + 5633 T^{4} - 21232 T^{5} - 326610 T^{6} + 250888 T^{7} + 16488036 T^{8} + 250888 p T^{9} - 326610 p^{2} T^{10} - 21232 p^{3} T^{11} + 5633 p^{4} T^{12} + 560 p^{5} T^{13} - 2 p^{7} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 18 T + 49 T^{2} - 78 T^{3} + 10413 T^{4} + 80892 T^{5} - 97690 T^{6} + 1307016 T^{7} + 34904282 T^{8} + 1307016 p T^{9} - 97690 p^{2} T^{10} + 80892 p^{3} T^{11} + 10413 p^{4} T^{12} - 78 p^{5} T^{13} + 49 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 6 T - 137 T^{2} + 382 T^{3} + 14003 T^{4} - 16208 T^{5} - 950024 T^{6} + 347280 T^{7} + 48885270 T^{8} + 347280 p T^{9} - 950024 p^{2} T^{10} - 16208 p^{3} T^{11} + 14003 p^{4} T^{12} + 382 p^{5} T^{13} - 137 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 10 T - 61 T^{2} + 298 T^{3} + 5825 T^{4} + 6964 T^{5} - 390522 T^{6} + 3760 p T^{7} + 12265818 T^{8} + 3760 p^{2} T^{9} - 390522 p^{2} T^{10} + 6964 p^{3} T^{11} + 5825 p^{4} T^{12} + 298 p^{5} T^{13} - 61 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 8 T - 158 T^{2} - 816 T^{3} + 19465 T^{4} + 52960 T^{5} - 1704654 T^{6} - 916232 T^{7} + 121857604 T^{8} - 916232 p T^{9} - 1704654 p^{2} T^{10} + 52960 p^{3} T^{11} + 19465 p^{4} T^{12} - 816 p^{5} T^{13} - 158 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 18 T + 17 T^{2} - 766 T^{3} + 7931 T^{4} + 88888 T^{5} - 387128 T^{6} + 222360 T^{7} + 60297142 T^{8} + 222360 p T^{9} - 387128 p^{2} T^{10} + 88888 p^{3} T^{11} + 7931 p^{4} T^{12} - 766 p^{5} T^{13} + 17 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 4 T - 102 T^{2} + 648 T^{3} - 407 T^{4} - 6872 T^{5} - 3042 p T^{6} - 1238100 T^{7} + 52149540 T^{8} - 1238100 p T^{9} - 3042 p^{3} T^{10} - 6872 p^{3} T^{11} - 407 p^{4} T^{12} + 648 p^{5} T^{13} - 102 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 6 T - 99 T^{2} - 1142 T^{3} + 13489 T^{4} + 102980 T^{5} + 685522 T^{6} - 8461384 T^{7} - 60614966 T^{8} - 8461384 p T^{9} + 685522 p^{2} T^{10} + 102980 p^{3} T^{11} + 13489 p^{4} T^{12} - 1142 p^{5} T^{13} - 99 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 194 T^{2} + 19617 T^{4} - 1428034 T^{6} + 95723876 T^{8} - 1428034 p^{2} T^{10} + 19617 p^{4} T^{12} - 194 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 + 14 T + T^{2} + 46 T^{3} + 6685 T^{4} + 16252 T^{5} + 303062 T^{6} - 561760 T^{7} - 67716270 T^{8} - 561760 p T^{9} + 303062 p^{2} T^{10} + 16252 p^{3} T^{11} + 6685 p^{4} T^{12} + 46 p^{5} T^{13} + p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 2 T + 255 T^{2} + 478 T^{3} + 29604 T^{4} + 478 p T^{5} + 255 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 2 T - 313 T^{2} - 610 T^{3} + 58717 T^{4} + 90420 T^{5} - 7549294 T^{6} - 3456704 T^{7} + 755010250 T^{8} - 3456704 p T^{9} - 7549294 p^{2} T^{10} + 90420 p^{3} T^{11} + 58717 p^{4} T^{12} - 610 p^{5} T^{13} - 313 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 12 T - 82 T^{2} - 2408 T^{3} - 5383 T^{4} + 209144 T^{5} + 2009614 T^{6} - 9879516 T^{7} - 279456620 T^{8} - 9879516 p T^{9} + 2009614 p^{2} T^{10} + 209144 p^{3} T^{11} - 5383 p^{4} T^{12} - 2408 p^{5} T^{13} - 82 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.18096703187735026508921143350, −4.14787229085487391418255641769, −3.87240972753823907720182930571, −3.69063806778335696100298226930, −3.65248162948557282318693581184, −3.51257378547856897771558067468, −3.48413169980041928593711717316, −3.10038391777312796510135150387, −3.02106323649187967972144661803, −2.89733049149527614292527194953, −2.85084389742012291569812459291, −2.69906903723037326278924557353, −2.60145044449763619023899942453, −2.43315907263584100692859734101, −2.13156623178939323535470286391, −1.95086608954312125358547173191, −1.88699990366628524136909843055, −1.78527629534676108434119946825, −1.73263307495946582746159028290, −1.40883564204202788736398537727, −1.40034248160446331180875069100, −1.02883684095398926844137555664, −0.893936413356933852554455562173, −0.75663905149875832281039025593, −0.02099577690925529988706107276, 0.02099577690925529988706107276, 0.75663905149875832281039025593, 0.893936413356933852554455562173, 1.02883684095398926844137555664, 1.40034248160446331180875069100, 1.40883564204202788736398537727, 1.73263307495946582746159028290, 1.78527629534676108434119946825, 1.88699990366628524136909843055, 1.95086608954312125358547173191, 2.13156623178939323535470286391, 2.43315907263584100692859734101, 2.60145044449763619023899942453, 2.69906903723037326278924557353, 2.85084389742012291569812459291, 2.89733049149527614292527194953, 3.02106323649187967972144661803, 3.10038391777312796510135150387, 3.48413169980041928593711717316, 3.51257378547856897771558067468, 3.65248162948557282318693581184, 3.69063806778335696100298226930, 3.87240972753823907720182930571, 4.14787229085487391418255641769, 4.18096703187735026508921143350
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.