Dirichlet series
| L(s) = 1 | − 28·7-s + 108·9-s + 4·17-s − 276·23-s + 600·25-s + 368·31-s + 944·47-s − 1.37e3·49-s − 3.02e3·63-s − 3.46e3·71-s + 296·73-s + 4.41e3·79-s + 5.34e3·81-s − 88·89-s − 4·97-s − 6.95e3·103-s + 668·113-s − 112·119-s + 7.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 432·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4·9-s + 0.0570·17-s − 2.50·23-s + 24/5·25-s + 2.13·31-s + 2.92·47-s − 3.99·49-s − 6.04·63-s − 5.79·71-s + 0.474·73-s + 6.28·79-s + 7.33·81-s − 0.104·89-s − 0.00418·97-s − 6.65·103-s + 0.556·113-s − 0.0862·119-s + 5.45·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.228·153-s + 0.000508·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{100}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.48127\times 10^{17}\) |
| Root analytic conductor: | \(7.77289\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{100} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.274593534\) |
| \(L(\frac12)\) | \(\approx\) | \(1.274593534\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p^{3} T^{2} + 6317 T^{4} - 275312 T^{6} + 3302590 p T^{8} - 295366792 T^{10} + 3302590 p^{7} T^{12} - 275312 p^{12} T^{14} + 6317 p^{18} T^{16} - 4 p^{27} T^{18} + p^{30} T^{20} \) |
| 5 | \( 1 - 24 p^{2} T^{2} + 191789 T^{4} - 42513504 T^{6} + 7224666922 T^{8} - 994659832592 T^{10} + 7224666922 p^{6} T^{12} - 42513504 p^{12} T^{14} + 191789 p^{18} T^{16} - 24 p^{26} T^{18} + p^{30} T^{20} \) | |
| 7 | \( ( 1 + 2 p T + 979 T^{2} + 5832 T^{3} + 372410 T^{4} + 662580 T^{5} + 372410 p^{3} T^{6} + 5832 p^{6} T^{7} + 979 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} )^{2} \) | |
| 11 | \( 1 - 60 p^{2} T^{2} + 24476093 T^{4} - 51189622576 T^{6} + 78543614693626 T^{8} - 105919424238644520 T^{10} + 78543614693626 p^{6} T^{12} - 51189622576 p^{12} T^{14} + 24476093 p^{18} T^{16} - 60 p^{26} T^{18} + p^{30} T^{20} \) | |
| 13 | \( 1 - 72 p^{2} T^{2} + 76341149 T^{4} - 324341675296 T^{6} + 1028186938895082 T^{8} - 2540163312339385904 T^{10} + 1028186938895082 p^{6} T^{12} - 324341675296 p^{12} T^{14} + 76341149 p^{18} T^{16} - 72 p^{26} T^{18} + p^{30} T^{20} \) | |
| 17 | \( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 98460610 p^{3} T^{6} - 102520 p^{6} T^{7} + 12653 p^{9} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 19 | \( 1 - 33948 T^{2} + 631492045 T^{4} - 8119820572464 T^{6} + 79259035180229178 T^{8} - \)\(60\!\cdots\!76\)\( T^{10} + 79259035180229178 p^{6} T^{12} - 8119820572464 p^{12} T^{14} + 631492045 p^{18} T^{16} - 33948 p^{24} T^{18} + p^{30} T^{20} \) | |
| 23 | \( ( 1 + 6 p T + 47715 T^{2} + 4085400 T^{3} + 882303386 T^{4} + 56970028764 T^{5} + 882303386 p^{3} T^{6} + 4085400 p^{6} T^{7} + 47715 p^{9} T^{8} + 6 p^{13} T^{9} + p^{15} T^{10} )^{2} \) | |
| 29 | \( 1 - 154168 T^{2} + 380663905 p T^{4} - 493373494152672 T^{6} + 16008852482124057194 T^{8} - \)\(42\!\cdots\!28\)\( T^{10} + 16008852482124057194 p^{6} T^{12} - 493373494152672 p^{12} T^{14} + 380663905 p^{19} T^{16} - 154168 p^{24} T^{18} + p^{30} T^{20} \) | |
| 31 | \( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 7638677322 p^{3} T^{6} - 19809056 p^{6} T^{7} + 134043 p^{9} T^{8} - 184 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 37 | \( 1 - 361064 T^{2} + 63572869101 T^{4} - 7160954906480288 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{10} + \)\(56\!\cdots\!06\)\( p^{6} T^{12} - 7160954906480288 p^{12} T^{14} + 63572869101 p^{18} T^{16} - 361064 p^{24} T^{18} + p^{30} T^{20} \) | |
| 41 | \( ( 1 + 220509 T^{2} - 10715136 T^{3} + 24270968490 T^{4} - 1235215122432 T^{5} + 24270968490 p^{3} T^{6} - 10715136 p^{6} T^{7} + 220509 p^{9} T^{8} + p^{15} T^{10} )^{2} \) | |
| 43 | \( 1 - 12836 p T^{2} + 145009588605 T^{4} - 24152295654927088 T^{6} + \)\(28\!\cdots\!30\)\( T^{8} - \)\(25\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!30\)\( p^{6} T^{12} - 24152295654927088 p^{12} T^{14} + 145009588605 p^{18} T^{16} - 12836 p^{25} T^{18} + p^{30} T^{20} \) | |
| 47 | \( ( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 90105579914 p^{3} T^{6} - 171516064 p^{6} T^{7} + 462219 p^{9} T^{8} - 472 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 53 | \( 1 - 525160 T^{2} + 112387404877 T^{4} - 12025532417207968 T^{6} + \)\(67\!\cdots\!30\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{10} + \)\(67\!\cdots\!30\)\( p^{6} T^{12} - 12025532417207968 p^{12} T^{14} + 112387404877 p^{18} T^{16} - 525160 p^{24} T^{18} + p^{30} T^{20} \) | |
| 59 | \( 1 - 958284 T^{2} + 532967592733 T^{4} - 207330855222256112 T^{6} + \)\(61\!\cdots\!62\)\( T^{8} - \)\(40\!\cdots\!88\)\( p^{2} T^{10} + \)\(61\!\cdots\!62\)\( p^{6} T^{12} - 207330855222256112 p^{12} T^{14} + 532967592733 p^{18} T^{16} - 958284 p^{24} T^{18} + p^{30} T^{20} \) | |
| 61 | \( 1 - 1146824 T^{2} + 8354214273 p T^{4} - 80080659050426912 T^{6} - \)\(14\!\cdots\!54\)\( T^{8} + \)\(80\!\cdots\!72\)\( T^{10} - \)\(14\!\cdots\!54\)\( p^{6} T^{12} - 80080659050426912 p^{12} T^{14} + 8354214273 p^{19} T^{16} - 1146824 p^{24} T^{18} + p^{30} T^{20} \) | |
| 67 | \( 1 - 2130652 T^{2} + 2127009110445 T^{4} - 1334413095892987440 T^{6} + \)\(59\!\cdots\!58\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{10} + \)\(59\!\cdots\!58\)\( p^{6} T^{12} - 1334413095892987440 p^{12} T^{14} + 2127009110445 p^{18} T^{16} - 2130652 p^{24} T^{18} + p^{30} T^{20} \) | |
| 71 | \( ( 1 + 1734 T + 2753587 T^{2} + 2611066280 T^{3} + 2272726706522 T^{4} + 1414434499369348 T^{5} + 2272726706522 p^{3} T^{6} + 2611066280 p^{6} T^{7} + 2753587 p^{9} T^{8} + 1734 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( ( 1 - 148 T + 1578093 T^{2} - 253463696 T^{3} + 1105903969594 T^{4} - 150341055768952 T^{5} + 1105903969594 p^{3} T^{6} - 253463696 p^{6} T^{7} + 1578093 p^{9} T^{8} - 148 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 79 | \( ( 1 - 2208 T + 3816107 T^{2} - 54694528 p T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 4245684014154 p^{3} T^{6} - 54694528 p^{7} T^{7} + 3816107 p^{9} T^{8} - 2208 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 2541516 T^{2} + 3317268679373 T^{4} - 33811498843796816 p T^{6} + \)\(18\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{10} + \)\(18\!\cdots\!66\)\( p^{6} T^{12} - 33811498843796816 p^{13} T^{14} + 3317268679373 p^{18} T^{16} - 2541516 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( ( 1 + 44 T + 1822557 T^{2} + 625587056 T^{3} + 1619388326906 T^{4} + 839474353817096 T^{5} + 1619388326906 p^{3} T^{6} + 625587056 p^{6} T^{7} + 1822557 p^{9} T^{8} + 44 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 97 | \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.12797331220385186908131125349, −3.02242931572872250490174557309, −2.97919269177880526235097329091, −2.84776555309735089414927666154, −2.82542231237696035842552273699, −2.66564811048867526520263158440, −2.52447101902318373505095020518, −2.45927271131856867061771147925, −2.14773733466127443854419669740, −2.08697124004289894218962419174, −2.03714013342576036549107376016, −1.77116347144717109233671762589, −1.76417115027676036234642027835, −1.74174066951988463662395439528, −1.42280062368013129917212251805, −1.32132562509647990123949139896, −1.16174484285238599294009784345, −1.15627354942952423732736285470, −1.09109862497464628654397917523, −0.78340091167248470916053479244, −0.72880544497162404231282097867, −0.64406394985347528742372269146, −0.51160288832866229769484533520, −0.10925737787235762421973151645, −0.080025516966570169121561703355, 0.080025516966570169121561703355, 0.10925737787235762421973151645, 0.51160288832866229769484533520, 0.64406394985347528742372269146, 0.72880544497162404231282097867, 0.78340091167248470916053479244, 1.09109862497464628654397917523, 1.15627354942952423732736285470, 1.16174484285238599294009784345, 1.32132562509647990123949139896, 1.42280062368013129917212251805, 1.74174066951988463662395439528, 1.76417115027676036234642027835, 1.77116347144717109233671762589, 2.03714013342576036549107376016, 2.08697124004289894218962419174, 2.14773733466127443854419669740, 2.45927271131856867061771147925, 2.52447101902318373505095020518, 2.66564811048867526520263158440, 2.82542231237696035842552273699, 2.84776555309735089414927666154, 2.97919269177880526235097329091, 3.02242931572872250490174557309, 3.12797331220385186908131125349
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.