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 − 4.79·25-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: | \(10\) |
| Selberg data: | \((20,\ 2^{100} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.70573473816378447599706886080, −3.60676758116657290243027709667, −3.51786857718362026378215359112, −3.50728346195588743414608377806, −3.45418286812149749746521593002, −2.99074238756984572827928866193, −2.98643133685656944717042867962, −2.91414073020808539088861518470, −2.87298082575040579154192564153, −2.71442606793662865383441168289, −2.68639000547097870119712695103, −2.47569206498056199220494659714, −2.46314318190157438604371392715, −2.41223286146436620098563852173, −2.37554241051811622092092400429, −1.98343232508317552865563173237, −1.72389915114416894867187576906, −1.72261729823941397109903884091, −1.67535015426031885301355290698, −1.58927744451896864184871354995, −1.44546828795848968364978880147, −1.26587496214482953470862746498, −1.21674533576091590028503796600, −1.06472253047039673708971502643, −0.934814839570824996018353715949, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.934814839570824996018353715949, 1.06472253047039673708971502643, 1.21674533576091590028503796600, 1.26587496214482953470862746498, 1.44546828795848968364978880147, 1.58927744451896864184871354995, 1.67535015426031885301355290698, 1.72261729823941397109903884091, 1.72389915114416894867187576906, 1.98343232508317552865563173237, 2.37554241051811622092092400429, 2.41223286146436620098563852173, 2.46314318190157438604371392715, 2.47569206498056199220494659714, 2.68639000547097870119712695103, 2.71442606793662865383441168289, 2.87298082575040579154192564153, 2.91414073020808539088861518470, 2.98643133685656944717042867962, 2.99074238756984572827928866193, 3.45418286812149749746521593002, 3.50728346195588743414608377806, 3.51786857718362026378215359112, 3.60676758116657290243027709667, 3.70573473816378447599706886080
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.