Dirichlet series
| L(s) = 1 | − 8·2-s + 32·4-s + 12·5-s − 8·7-s − 80·8-s − 96·10-s − 40·11-s − 24·13-s + 64·14-s + 120·16-s + 8·17-s + 384·20-s + 320·22-s − 16·23-s + 80·25-s + 192·26-s − 256·28-s − 64·31-s − 32·32-s − 64·34-s − 96·35-s − 24·37-s − 960·40-s + 56·41-s − 72·43-s − 1.28e3·44-s + 128·46-s + ⋯ |
| L(s) = 1 | − 4·2-s + 8·4-s + 12/5·5-s − 8/7·7-s − 10·8-s − 9.59·10-s − 3.63·11-s − 1.84·13-s + 32/7·14-s + 15/2·16-s + 8/17·17-s + 96/5·20-s + 14.5·22-s − 0.695·23-s + 16/5·25-s + 7.38·26-s − 9.14·28-s − 2.06·31-s − 32-s − 1.88·34-s − 2.74·35-s − 0.648·37-s − 24·40-s + 1.36·41-s − 1.67·43-s − 29.0·44-s + 2.78·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{24} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.58203\times 10^{6}\) |
| Root analytic conductor: | \(2.71237\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{24} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.2748455110\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2748455110\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 12 T + 64 T^{2} - 108 T^{3} - 24 p T^{4} - 108 p^{2} T^{5} + 64 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 7 | \( 1 + 8 T + 32 T^{2} + 256 T^{3} - 3442 T^{4} - 27464 T^{5} - 76800 T^{6} - 207864 T^{7} + 5077843 T^{8} - 207864 p^{2} T^{9} - 76800 p^{4} T^{10} - 27464 p^{6} T^{11} - 3442 p^{8} T^{12} + 256 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 20 T + 400 T^{2} + 4340 T^{3} + 55768 T^{4} + 4340 p^{2} T^{5} + 400 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 13 | \( 1 + 24 T + 288 T^{2} + 5856 T^{3} + 3622 p T^{4} - 417240 T^{5} - 6428160 T^{6} - 165397320 T^{7} - 3701745165 T^{8} - 165397320 p^{2} T^{9} - 6428160 p^{4} T^{10} - 417240 p^{6} T^{11} + 3622 p^{9} T^{12} + 5856 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 8 T + 32 T^{2} + 7832 T^{3} - 71596 T^{4} - 403384 T^{5} + 36188256 T^{6} + 64357608 T^{7} + 158386534 T^{8} + 64357608 p^{2} T^{9} + 36188256 p^{4} T^{10} - 403384 p^{6} T^{11} - 71596 p^{8} T^{12} + 7832 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 580 T^{2} + 475930 T^{4} - 192894064 T^{6} + 91634293315 T^{8} - 192894064 p^{4} T^{10} + 475930 p^{8} T^{12} - 580 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 + 16 T + 128 T^{2} + 10040 T^{3} + 228752 T^{4} - 2307400 T^{5} - 15797856 T^{6} - 601862832 T^{7} - 3438620990 T^{8} - 601862832 p^{2} T^{9} - 15797856 p^{4} T^{10} - 2307400 p^{6} T^{11} + 228752 p^{8} T^{12} + 10040 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 3760 T^{2} + 7513360 T^{4} - 10124362384 T^{6} + 9927642064450 T^{8} - 10124362384 p^{4} T^{10} + 7513360 p^{8} T^{12} - 3760 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 + 32 T + 1384 T^{2} - 19744 T^{3} - 216494 T^{4} - 19744 p^{2} T^{5} + 1384 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 + 24 T + 288 T^{2} + 62016 T^{3} - 1193810 T^{4} - 54832824 T^{5} + 950821632 T^{6} - 7540071336 T^{7} + 489879159603 T^{8} - 7540071336 p^{2} T^{9} + 950821632 p^{4} T^{10} - 54832824 p^{6} T^{11} - 1193810 p^{8} T^{12} + 62016 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 - 28 T + 4480 T^{2} - 2372 p T^{3} + 9455128 T^{4} - 2372 p^{3} T^{5} + 4480 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 + 72 T + 2592 T^{2} + 39960 T^{3} + 5556412 T^{4} + 548106696 T^{5} + 25859863008 T^{6} + 705930936408 T^{7} + 15548175298374 T^{8} + 705930936408 p^{2} T^{9} + 25859863008 p^{4} T^{10} + 548106696 p^{6} T^{11} + 5556412 p^{8} T^{12} + 39960 p^{10} T^{13} + 2592 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 124416 T^{3} + 6072004 T^{4} + 373248 p T^{5} + 7739670528 T^{6} + 680045663232 T^{7} + 6691298905734 T^{8} + 680045663232 p^{2} T^{9} + 7739670528 p^{4} T^{10} + 373248 p^{7} T^{11} + 6072004 p^{8} T^{12} + 124416 p^{10} T^{13} + p^{16} T^{16} \) | |
| 53 | \( 1 + 80 T + 3200 T^{2} + 292120 T^{3} + 10074896 T^{4} - 402405800 T^{5} - 21765084000 T^{6} - 2335299106800 T^{7} - 229685261539070 T^{8} - 2335299106800 p^{2} T^{9} - 21765084000 p^{4} T^{10} - 402405800 p^{6} T^{11} + 10074896 p^{8} T^{12} + 292120 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 14008 T^{2} + 100975756 T^{4} - 492995752648 T^{6} + 1888076779441894 T^{8} - 492995752648 p^{4} T^{10} + 100975756 p^{8} T^{12} - 14008 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 108 T + 12490 T^{2} - 969552 T^{3} + 68045523 T^{4} - 969552 p^{2} T^{5} + 12490 p^{4} T^{6} - 108 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 - 136 T + 9248 T^{2} - 705536 T^{3} + 33707342 T^{4} + 897952072 T^{5} - 184956456960 T^{6} + 22832526785592 T^{7} - 2128448215323821 T^{8} + 22832526785592 p^{2} T^{9} - 184956456960 p^{4} T^{10} + 897952072 p^{6} T^{11} + 33707342 p^{8} T^{12} - 705536 p^{10} T^{13} + 9248 p^{12} T^{14} - 136 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 28 T + 16048 T^{2} + 231316 T^{3} + 109038184 T^{4} + 231316 p^{2} T^{5} + 16048 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 368 T + 67712 T^{2} - 8839312 T^{3} + 943013726 T^{4} - 87425766400 T^{5} + 7386054936960 T^{6} - 587580358449120 T^{7} + 44199309763992355 T^{8} - 587580358449120 p^{2} T^{9} + 7386054936960 p^{4} T^{10} - 87425766400 p^{6} T^{11} + 943013726 p^{8} T^{12} - 8839312 p^{10} T^{13} + 67712 p^{12} T^{14} - 368 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 25340 T^{2} + 370619530 T^{4} - 3658708310384 T^{6} + 26467945053296275 T^{8} - 3658708310384 p^{4} T^{10} + 370619530 p^{8} T^{12} - 25340 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 - 168 T + 14112 T^{2} - 916800 T^{3} - 16348688 T^{4} + 4879673856 T^{5} - 168811402752 T^{6} - 16030525612392 T^{7} + 3571925738115714 T^{8} - 16030525612392 p^{2} T^{9} - 168811402752 p^{4} T^{10} + 4879673856 p^{6} T^{11} - 16348688 p^{8} T^{12} - 916800 p^{10} T^{13} + 14112 p^{12} T^{14} - 168 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 34744 T^{2} + 427201228 T^{4} - 1863800468296 T^{6} + 2580261126216742 T^{8} - 1863800468296 p^{4} T^{10} + 427201228 p^{8} T^{12} - 34744 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 + 312 T + 48672 T^{2} + 6967680 T^{3} + 1062851086 T^{4} + 133636973544 T^{5} + 14237929979136 T^{6} + 1553115219249528 T^{7} + 162662200396802835 T^{8} + 1553115219249528 p^{2} T^{9} + 14237929979136 p^{4} T^{10} + 133636973544 p^{6} T^{11} + 1062851086 p^{8} T^{12} + 6967680 p^{10} T^{13} + 48672 p^{12} T^{14} + 312 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−5.22484041655693816929755836648, −5.21163872928654980305866265414, −4.76584703474072910500199190211, −4.75543939025247428031016259734, −4.69657934892309397250991694702, −4.50792331068814571894094624087, −4.20038646135608457859117084791, −3.78046633505642825787948939473, −3.69857492196017408664227125299, −3.53334088993581534931794482628, −3.33220308286275044153637287016, −3.18139463178982154775879454087, −2.74639128598358044864235028200, −2.66993606969624752926930020673, −2.61131573476276352897702861920, −2.34437271974939728226819702891, −2.15557169184256291185029624561, −2.02281418403474007643836066484, −2.01000971734080356047765334173, −1.59608802808158665060268182157, −1.52903910314353430013784854878, −0.826156839722551317834885956186, −0.67161685326983419943729563936, −0.40016101647461352776988988529, −0.30464754717572462909757076254, 0.30464754717572462909757076254, 0.40016101647461352776988988529, 0.67161685326983419943729563936, 0.826156839722551317834885956186, 1.52903910314353430013784854878, 1.59608802808158665060268182157, 2.01000971734080356047765334173, 2.02281418403474007643836066484, 2.15557169184256291185029624561, 2.34437271974939728226819702891, 2.61131573476276352897702861920, 2.66993606969624752926930020673, 2.74639128598358044864235028200, 3.18139463178982154775879454087, 3.33220308286275044153637287016, 3.53334088993581534931794482628, 3.69857492196017408664227125299, 3.78046633505642825787948939473, 4.20038646135608457859117084791, 4.50792331068814571894094624087, 4.69657934892309397250991694702, 4.75543939025247428031016259734, 4.76584703474072910500199190211, 5.21163872928654980305866265414, 5.22484041655693816929755836648
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.