Dirichlet series
| L(s) = 1 | + 4·5-s − 8·8-s − 4·13-s + 5·16-s − 8·17-s − 8·23-s + 4·25-s + 32·29-s + 4·37-s − 32·40-s + 40·43-s − 24·47-s + 40·53-s + 80·59-s − 32·61-s + 32·64-s − 16·65-s − 48·67-s − 20·73-s + 20·80-s − 24·83-s − 32·85-s + 56·89-s + 12·97-s − 8·103-s + 32·104-s − 88·107-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.82·8-s − 1.10·13-s + 5/4·16-s − 1.94·17-s − 1.66·23-s + 4/5·25-s + 5.94·29-s + 0.657·37-s − 5.05·40-s + 6.09·43-s − 3.50·47-s + 5.49·53-s + 10.4·59-s − 4.09·61-s + 4·64-s − 1.98·65-s − 5.86·67-s − 2.34·73-s + 2.23·80-s − 2.63·83-s − 3.47·85-s + 5.93·89-s + 1.21·97-s − 0.788·103-s + 3.13·104-s − 8.50·107-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(64128.6\) |
| Root analytic conductor: | \(1.58596\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.628329370\) |
| \(L(\frac12)\) | \(\approx\) | \(5.628329370\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 4 T + 12 T^{2} - 12 T^{3} - p T^{4} + 32 p T^{5} - 408 T^{6} + 32 p^{2} T^{7} - p^{3} T^{8} - 12 p^{3} T^{9} + 12 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 7 | \( ( 1 + T^{4} )^{3} \) | |
| good | 2 | \( 1 + p^{3} T^{3} - 5 T^{4} + p^{5} T^{6} - p^{5} T^{7} - 5 T^{8} + 5 p^{4} T^{9} - 3 p^{5} T^{10} - 9 p^{3} T^{11} + 193 T^{12} - 9 p^{4} T^{13} - 3 p^{7} T^{14} + 5 p^{7} T^{15} - 5 p^{4} T^{16} - p^{10} T^{17} + p^{11} T^{18} - 5 p^{8} T^{20} + p^{12} T^{21} + p^{12} T^{24} \) |
| 11 | \( 1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 6805160 p^{2} T^{14} + 494911 p^{4} T^{16} - 30084 p^{6} T^{18} + 1458 p^{8} T^{20} - 52 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 4 T + 8 T^{2} + 108 T^{3} + 298 T^{4} + 204 T^{5} + 328 p T^{6} + 17924 T^{7} + 42687 T^{8} + 200904 T^{9} + 1148048 T^{10} + 3595224 T^{11} + 7337420 T^{12} + 3595224 p T^{13} + 1148048 p^{2} T^{14} + 200904 p^{3} T^{15} + 42687 p^{4} T^{16} + 17924 p^{5} T^{17} + 328 p^{7} T^{18} + 204 p^{7} T^{19} + 298 p^{8} T^{20} + 108 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 8 T + 32 T^{2} + 232 T^{3} + 1238 T^{4} + 3432 T^{5} + 14752 T^{6} + 67976 T^{7} + 254479 T^{8} + 1447664 T^{9} + 6834752 T^{10} + 33133872 T^{11} + 165039860 T^{12} + 33133872 p T^{13} + 6834752 p^{2} T^{14} + 1447664 p^{3} T^{15} + 254479 p^{4} T^{16} + 67976 p^{5} T^{17} + 14752 p^{6} T^{18} + 3432 p^{7} T^{19} + 1238 p^{8} T^{20} + 232 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 138383432 p^{2} T^{14} + 5560223 p^{4} T^{16} - 179332 p^{6} T^{18} + 4642 p^{8} T^{20} - 84 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + 518 T^{4} - 1752 T^{5} - 13664 T^{6} - 145640 T^{7} - 745937 T^{8} + 382288 T^{9} + 8743232 T^{10} + 77776560 T^{11} + 600339476 T^{12} + 77776560 p T^{13} + 8743232 p^{2} T^{14} + 382288 p^{3} T^{15} - 745937 p^{4} T^{16} - 145640 p^{5} T^{17} - 13664 p^{6} T^{18} - 1752 p^{7} T^{19} + 518 p^{8} T^{20} + 8 p^{10} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 - 16 T + 216 T^{2} - 1904 T^{3} + 15483 T^{4} - 3392 p T^{5} + 589488 T^{6} - 3392 p^{2} T^{7} + 15483 p^{2} T^{8} - 1904 p^{3} T^{9} + 216 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 10856 p T^{7} + 6583 p^{2} T^{8} - 168 p^{3} T^{9} + 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 4 T + 8 T^{2} - 28 T^{3} + 122 T^{4} - 124 T^{5} - 88 T^{6} + 102780 T^{7} - 827937 T^{8} + 2005016 T^{9} - 4255984 T^{10} + 201457960 T^{11} - 3018818132 T^{12} + 201457960 p T^{13} - 4255984 p^{2} T^{14} + 2005016 p^{3} T^{15} - 827937 p^{4} T^{16} + 102780 p^{5} T^{17} - 88 p^{6} T^{18} - 124 p^{7} T^{19} + 122 p^{8} T^{20} - 28 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 11090030832 p^{2} T^{14} + 198407727 p^{4} T^{16} - 2820520 p^{6} T^{18} + 30646 p^{8} T^{20} - 232 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 40 T + 800 T^{2} - 11608 T^{3} + 144630 T^{4} - 1609192 T^{5} + 16036512 T^{6} - 146487000 T^{7} + 1250458847 T^{8} - 9986195856 T^{9} + 74890568512 T^{10} - 532487979120 T^{11} + 3591291202036 T^{12} - 532487979120 p T^{13} + 74890568512 p^{2} T^{14} - 9986195856 p^{3} T^{15} + 1250458847 p^{4} T^{16} - 146487000 p^{5} T^{17} + 16036512 p^{6} T^{18} - 1609192 p^{7} T^{19} + 144630 p^{8} T^{20} - 11608 p^{9} T^{21} + 800 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 30374 T^{4} + 283512 T^{5} + 2319776 T^{6} + 18683912 T^{7} + 148359759 T^{8} + 1083670000 T^{9} + 7566143296 T^{10} + 53830728976 T^{11} + 378456487636 T^{12} + 53830728976 p T^{13} + 7566143296 p^{2} T^{14} + 1083670000 p^{3} T^{15} + 148359759 p^{4} T^{16} + 18683912 p^{5} T^{17} + 2319776 p^{6} T^{18} + 283512 p^{7} T^{19} + 30374 p^{8} T^{20} + 2920 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 40 T + 800 T^{2} - 11880 T^{3} + 147222 T^{4} - 1477736 T^{5} + 11899040 T^{6} - 74456936 T^{7} + 249326879 T^{8} + 1557249200 T^{9} - 37151695552 T^{10} + 419908594096 T^{11} - 3501062916044 T^{12} + 419908594096 p T^{13} - 37151695552 p^{2} T^{14} + 1557249200 p^{3} T^{15} + 249326879 p^{4} T^{16} - 74456936 p^{5} T^{17} + 11899040 p^{6} T^{18} - 1477736 p^{7} T^{19} + 147222 p^{8} T^{20} - 11880 p^{9} T^{21} + 800 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 40 T + 986 T^{2} - 16824 T^{3} + 222455 T^{4} - 2325552 T^{5} + 19830604 T^{6} - 2325552 p T^{7} + 222455 p^{2} T^{8} - 16824 p^{3} T^{9} + 986 p^{4} T^{10} - 40 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 270400 p T^{7} + 31591 p^{2} T^{8} + 2800 p^{3} T^{9} + 254 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 + 48 T + 1152 T^{2} + 18992 T^{3} + 248566 T^{4} + 2816976 T^{5} + 29214848 T^{6} + 287145808 T^{7} + 2747657919 T^{8} + 25947593440 T^{9} + 239514732288 T^{10} + 2122341535712 T^{11} + 17856126120308 T^{12} + 2122341535712 p T^{13} + 239514732288 p^{2} T^{14} + 25947593440 p^{3} T^{15} + 2747657919 p^{4} T^{16} + 287145808 p^{5} T^{17} + 29214848 p^{6} T^{18} + 2816976 p^{7} T^{19} + 248566 p^{8} T^{20} + 18992 p^{9} T^{21} + 1152 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 25299911144 p^{2} T^{14} + 571593487 p^{4} T^{16} - 8307268 p^{6} T^{18} + 75490 p^{8} T^{20} - 404 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 20 T + 200 T^{2} + 1452 T^{3} + 7482 T^{4} + 35804 T^{5} + 273832 T^{6} + 2000484 T^{7} + 27883951 T^{8} + 188107816 T^{9} - 59632 p^{2} T^{10} - 470201768 p T^{11} - 473910414484 T^{12} - 470201768 p^{2} T^{13} - 59632 p^{4} T^{14} + 188107816 p^{3} T^{15} + 27883951 p^{4} T^{16} + 2000484 p^{5} T^{17} + 273832 p^{6} T^{18} + 35804 p^{7} T^{19} + 7482 p^{8} T^{20} + 1452 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 90900376040 p^{2} T^{14} + 1031952175 p^{4} T^{16} - 9878564 p^{6} T^{18} + 73090 p^{8} T^{20} - 372 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 + 24 T + 288 T^{2} + 2760 T^{3} + 26518 T^{4} + 273208 T^{5} + 2728608 T^{6} + 24159272 T^{7} + 240213375 T^{8} + 2803489520 T^{9} + 29745965888 T^{10} + 283235701200 T^{11} + 2619914616372 T^{12} + 283235701200 p T^{13} + 29745965888 p^{2} T^{14} + 2803489520 p^{3} T^{15} + 240213375 p^{4} T^{16} + 24159272 p^{5} T^{17} + 2728608 p^{6} T^{18} + 273208 p^{7} T^{19} + 26518 p^{8} T^{20} + 2760 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 28 T + 684 T^{2} - 11188 T^{3} + 161827 T^{4} - 1876496 T^{5} + 19427016 T^{6} - 1876496 p T^{7} + 161827 p^{2} T^{8} - 11188 p^{3} T^{9} + 684 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 12 T + 72 T^{2} - 884 T^{3} + 15450 T^{4} - 102628 T^{5} + 509864 T^{6} - 1529948 T^{7} - 62272049 T^{8} + 410787880 T^{9} - 1704538608 T^{10} + 69640900504 T^{11} - 1520682376276 T^{12} + 69640900504 p T^{13} - 1704538608 p^{2} T^{14} + 410787880 p^{3} T^{15} - 62272049 p^{4} T^{16} - 1529948 p^{5} T^{17} + 509864 p^{6} T^{18} - 102628 p^{7} T^{19} + 15450 p^{8} T^{20} - 884 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.93209937871713419964549077761, −3.89263552595831133750671769977, −3.84595622880058365022689380441, −3.84008012048766561964907122564, −3.47361494193605769557961373381, −3.45095396170603426559265730050, −3.14342578009131277519658723100, −2.98251398896435187621302298518, −2.88171366013468667763269588187, −2.73808679088739839102645917657, −2.72598295437535964102393127303, −2.72323225185120017843992028112, −2.66701522798562291360951640618, −2.66363127018185519573059355125, −2.41309186144013700810469577513, −2.18928620292958625319639485838, −2.05053711979531005055209816864, −1.86029513879543281851376968163, −1.82537741977140221309124362915, −1.66074386388108311923597422288, −1.25415556064352565915564887791, −1.08087976811984294688981498893, −0.66436259923093449259876878491, −0.64638391893266245405219369214, −0.61889192115712973724395235383, 0.61889192115712973724395235383, 0.64638391893266245405219369214, 0.66436259923093449259876878491, 1.08087976811984294688981498893, 1.25415556064352565915564887791, 1.66074386388108311923597422288, 1.82537741977140221309124362915, 1.86029513879543281851376968163, 2.05053711979531005055209816864, 2.18928620292958625319639485838, 2.41309186144013700810469577513, 2.66363127018185519573059355125, 2.66701522798562291360951640618, 2.72323225185120017843992028112, 2.72598295437535964102393127303, 2.73808679088739839102645917657, 2.88171366013468667763269588187, 2.98251398896435187621302298518, 3.14342578009131277519658723100, 3.45095396170603426559265730050, 3.47361494193605769557961373381, 3.84008012048766561964907122564, 3.84595622880058365022689380441, 3.89263552595831133750671769977, 3.93209937871713419964549077761
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.