Dirichlet series
| L(s) = 1 | + 4·7-s − 32·11-s + 4·13-s − 52·17-s − 40·23-s − 96·31-s + 60·37-s − 152·41-s − 88·43-s − 16·47-s + 8·49-s − 108·53-s + 264·61-s − 216·67-s + 240·71-s + 208·73-s − 128·77-s − 18·81-s + 336·83-s + 16·91-s + 208·97-s − 400·101-s − 260·103-s − 272·107-s + 132·113-s − 208·119-s + 516·121-s + ⋯ |
| L(s) = 1 | + 4/7·7-s − 2.90·11-s + 4/13·13-s − 3.05·17-s − 1.73·23-s − 3.09·31-s + 1.62·37-s − 3.70·41-s − 2.04·43-s − 0.340·47-s + 8/49·49-s − 2.03·53-s + 4.32·61-s − 3.22·67-s + 3.38·71-s + 2.84·73-s − 1.66·77-s − 2/9·81-s + 4.04·83-s + 0.175·91-s + 2.14·97-s − 3.96·101-s − 2.52·103-s − 2.54·107-s + 1.16·113-s − 1.74·119-s + 4.26·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.30656\times 10^{12}\) |
| Root analytic conductor: | \(5.71818\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.2866821060\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2866821060\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + 8 T^{2} - 316 T^{3} - 2816 T^{4} + 2108 T^{5} + 64024 T^{6} - 120828 T^{7} + 11331966 T^{8} - 120828 p^{2} T^{9} + 64024 p^{4} T^{10} + 2108 p^{6} T^{11} - 2816 p^{8} T^{12} - 316 p^{10} T^{13} + 8 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 16 T + 126 T^{2} - 1888 T^{3} - 31078 T^{4} - 1888 p^{2} T^{5} + 126 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 13 | \( 1 - 4 T + 8 T^{2} - 124 p T^{3} + 1072 p T^{4} + 328076 T^{5} - 124520 T^{6} + 28503876 T^{7} - 763526370 T^{8} + 28503876 p^{2} T^{9} - 124520 p^{4} T^{10} + 328076 p^{6} T^{11} + 1072 p^{9} T^{12} - 124 p^{11} T^{13} + 8 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 52 T + 1352 T^{2} + 28204 T^{3} + 490768 T^{4} + 6510964 T^{5} + 72784600 T^{6} + 481409004 T^{7} + 154737438 T^{8} + 481409004 p^{2} T^{9} + 72784600 p^{4} T^{10} + 6510964 p^{6} T^{11} + 490768 p^{8} T^{12} + 28204 p^{10} T^{13} + 1352 p^{12} T^{14} + 52 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 2000 T^{2} + 1889820 T^{4} - 1128357424 T^{6} + 476406585350 T^{8} - 1128357424 p^{4} T^{10} + 1889820 p^{8} T^{12} - 2000 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 + 40 T + 800 T^{2} + 18248 T^{3} - 19516 T^{4} - 7340008 T^{5} - 111492768 T^{6} - 1524968904 T^{7} - 5167201274 T^{8} - 1524968904 p^{2} T^{9} - 111492768 p^{4} T^{10} - 7340008 p^{6} T^{11} - 19516 p^{8} T^{12} + 18248 p^{10} T^{13} + 800 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 2228 T^{2} + 1857576 T^{4} - 965861788 T^{6} + 647582808974 T^{8} - 965861788 p^{4} T^{10} + 1857576 p^{8} T^{12} - 2228 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 + 48 T + 1740 T^{2} - 1968 T^{3} - 318442 T^{4} - 1968 p^{2} T^{5} + 1740 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 - 60 T + 1800 T^{2} + 32652 T^{3} - 6760976 T^{4} + 249442548 T^{5} - 2263719528 T^{6} - 292468480836 T^{7} + 19178941117086 T^{8} - 292468480836 p^{2} T^{9} - 2263719528 p^{4} T^{10} + 249442548 p^{6} T^{11} - 6760976 p^{8} T^{12} + 32652 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 + 76 T + 5776 T^{2} + 267748 T^{3} + 13703518 T^{4} + 267748 p^{2} T^{5} + 5776 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 + 88 T + 3872 T^{2} + 100696 T^{3} + 1714468 T^{4} + 169137736 T^{5} + 13315542880 T^{6} + 642392563656 T^{7} + 29742475153158 T^{8} + 642392563656 p^{2} T^{9} + 13315542880 p^{4} T^{10} + 169137736 p^{6} T^{11} + 1714468 p^{8} T^{12} + 100696 p^{10} T^{13} + 3872 p^{12} T^{14} + 88 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 16 T + 128 T^{2} - 128816 T^{3} + 2412964 T^{4} + 504683248 T^{5} + 16062853504 T^{6} + 177987325872 T^{7} - 50194759450554 T^{8} + 177987325872 p^{2} T^{9} + 16062853504 p^{4} T^{10} + 504683248 p^{6} T^{11} + 2412964 p^{8} T^{12} - 128816 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 108 T + 5832 T^{2} + 412884 T^{3} + 23090224 T^{4} + 11383740 p T^{5} + 15734940120 T^{6} + 223612868052 T^{7} - 24915462238434 T^{8} + 223612868052 p^{2} T^{9} + 15734940120 p^{4} T^{10} + 11383740 p^{7} T^{11} + 23090224 p^{8} T^{12} + 412884 p^{10} T^{13} + 5832 p^{12} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 25500 T^{2} + 290547640 T^{4} - 1939529752884 T^{6} + 8319211197765870 T^{8} - 1939529752884 p^{4} T^{10} + 290547640 p^{8} T^{12} - 25500 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 132 T + 13872 T^{2} - 941868 T^{3} + 61614014 T^{4} - 941868 p^{2} T^{5} + 13872 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 + 216 T + 23328 T^{2} + 2220888 T^{3} + 213731236 T^{4} + 17658282696 T^{5} + 1294438543200 T^{6} + 97283636525256 T^{7} + 7010705853400710 T^{8} + 97283636525256 p^{2} T^{9} + 1294438543200 p^{4} T^{10} + 17658282696 p^{6} T^{11} + 213731236 p^{8} T^{12} + 2220888 p^{10} T^{13} + 23328 p^{12} T^{14} + 216 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 - 120 T + 20340 T^{2} - 1543656 T^{3} + 149872550 T^{4} - 1543656 p^{2} T^{5} + 20340 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 208 T + 21632 T^{2} - 2121136 T^{3} + 133201276 T^{4} - 248473360 T^{5} - 580118578304 T^{6} + 88473819716496 T^{7} - 8873634681591930 T^{8} + 88473819716496 p^{2} T^{9} - 580118578304 p^{4} T^{10} - 248473360 p^{6} T^{11} + 133201276 p^{8} T^{12} - 2121136 p^{10} T^{13} + 21632 p^{12} T^{14} - 208 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 32504 T^{2} + 462697980 T^{4} - 4035012008776 T^{6} + 27090604999745414 T^{8} - 4035012008776 p^{4} T^{10} + 462697980 p^{8} T^{12} - 32504 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 - 336 T + 56448 T^{2} - 7845648 T^{3} + 1037744740 T^{4} - 116459238960 T^{5} + 11328785476992 T^{6} - 1064762407254768 T^{7} + 93992795964479238 T^{8} - 1064762407254768 p^{2} T^{9} + 11328785476992 p^{4} T^{10} - 116459238960 p^{6} T^{11} + 1037744740 p^{8} T^{12} - 7845648 p^{10} T^{13} + 56448 p^{12} T^{14} - 336 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 25856 T^{2} + 291349884 T^{4} - 1811616834304 T^{6} + 9967156810684550 T^{8} - 1811616834304 p^{4} T^{10} + 291349884 p^{8} T^{12} - 25856 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 - 208 T + 21632 T^{2} - 956336 T^{3} - 203065348 T^{4} + 29702194672 T^{5} - 1328057611392 T^{6} - 139759626438000 T^{7} + 32182715530229254 T^{8} - 139759626438000 p^{2} T^{9} - 1328057611392 p^{4} T^{10} + 29702194672 p^{6} T^{11} - 203065348 p^{8} T^{12} - 956336 p^{10} T^{13} + 21632 p^{12} T^{14} - 208 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−3.94420923422537470305099584977, −3.78161436355158805027030905304, −3.60599904485952279068392794343, −3.52605076462191525483314348681, −3.52323457159174052547869613625, −3.43174218108118931657737290217, −3.07460112527400656716140940266, −2.95541534172617621610148362977, −2.88416794057919398334695423649, −2.77594368326928209134043439895, −2.29761267549241239004906712523, −2.28892103651475183260095858056, −2.27990845187580873432009438712, −2.23027452526585157698022246993, −2.09701606936515163278855740889, −1.92801912946765862497774348006, −1.78434199304439915270339616005, −1.63989444014911121086235730723, −1.22270450982872365102780179348, −1.16262499385217978828501617158, −1.14277843696561215107834929528, −0.44166232721552447955394099363, −0.41214445570416227748778875565, −0.29200503110419599187992079312, −0.087310701433184087008261938190, 0.087310701433184087008261938190, 0.29200503110419599187992079312, 0.41214445570416227748778875565, 0.44166232721552447955394099363, 1.14277843696561215107834929528, 1.16262499385217978828501617158, 1.22270450982872365102780179348, 1.63989444014911121086235730723, 1.78434199304439915270339616005, 1.92801912946765862497774348006, 2.09701606936515163278855740889, 2.23027452526585157698022246993, 2.27990845187580873432009438712, 2.28892103651475183260095858056, 2.29761267549241239004906712523, 2.77594368326928209134043439895, 2.88416794057919398334695423649, 2.95541534172617621610148362977, 3.07460112527400656716140940266, 3.43174218108118931657737290217, 3.52323457159174052547869613625, 3.52605076462191525483314348681, 3.60599904485952279068392794343, 3.78161436355158805027030905304, 3.94420923422537470305099584977
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.