Dirichlet series
| L(s) = 1 | − 18·9-s − 80·17-s + 40·29-s − 320·37-s + 136·41-s + 188·49-s + 176·53-s + 40·61-s − 448·73-s + 189·81-s + 8·89-s − 224·97-s − 520·101-s + 280·109-s − 48·113-s + 860·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.44e3·153-s + 157-s + 163-s + 167-s − 860·169-s + ⋯ |
| L(s) = 1 | − 2·9-s − 4.70·17-s + 1.37·29-s − 8.64·37-s + 3.31·41-s + 3.83·49-s + 3.32·53-s + 0.655·61-s − 6.13·73-s + 7/3·81-s + 8/89·89-s − 2.30·97-s − 5.14·101-s + 2.56·109-s − 0.424·113-s + 7.10·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 9.41·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.08·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 3^{12} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.11724\times 10^{21}\) |
| Root analytic conductor: | \(8.08673\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 3^{12} \cdot 5^{24} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.3472413626\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3472413626\) |
| \(L(2)\) | 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 | \( ( 1 + p T^{2} )^{6} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 188 T^{2} + 2990 p T^{4} - 223828 p T^{6} + 90881327 T^{8} - 4463729016 T^{10} + 214353683868 T^{12} - 4463729016 p^{4} T^{14} + 90881327 p^{8} T^{16} - 223828 p^{13} T^{18} + 2990 p^{17} T^{20} - 188 p^{20} T^{22} + p^{24} T^{24} \) |
| 11 | \( 1 - 860 T^{2} + 327906 T^{4} - 72080428 T^{6} + 10020403343 T^{8} - 971292022200 T^{10} + 94781500761820 T^{12} - 971292022200 p^{4} T^{14} + 10020403343 p^{8} T^{16} - 72080428 p^{12} T^{18} + 327906 p^{16} T^{20} - 860 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 430 T^{2} - 160 p T^{3} + 125999 T^{4} - 493280 T^{5} + 2049524 p T^{6} - 493280 p^{2} T^{7} + 125999 p^{4} T^{8} - 160 p^{7} T^{9} + 430 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 + 40 T + 1758 T^{2} + 47912 T^{3} + 1245311 T^{4} + 25289136 T^{5} + 475806148 T^{6} + 25289136 p^{2} T^{7} + 1245311 p^{4} T^{8} + 47912 p^{6} T^{9} + 1758 p^{8} T^{10} + 40 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 - 2924 T^{2} + 4209506 T^{4} - 3946403452 T^{6} + 2683968573647 T^{8} - 1394336750927064 T^{10} + 566929155961181916 T^{12} - 1394336750927064 p^{4} T^{14} + 2683968573647 p^{8} T^{16} - 3946403452 p^{12} T^{18} + 4209506 p^{16} T^{20} - 2924 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 - 2540 T^{2} + 3838146 T^{4} - 4073305276 T^{6} + 146639294761 p T^{8} - 2290088790769368 T^{10} + 1311251066381582236 T^{12} - 2290088790769368 p^{4} T^{14} + 146639294761 p^{9} T^{16} - 4073305276 p^{12} T^{18} + 3838146 p^{16} T^{20} - 2540 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 20 T + 2458 T^{2} - 33188 T^{3} + 3302351 T^{4} - 47144360 T^{5} + 3441947084 T^{6} - 47144360 p^{2} T^{7} + 3302351 p^{4} T^{8} - 33188 p^{6} T^{9} + 2458 p^{8} T^{10} - 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 4972 T^{2} + 10961922 T^{4} - 13437805244 T^{6} + 8217916337135 T^{8} + 1042318389291048 T^{10} - 5452908842588815844 T^{12} + 1042318389291048 p^{4} T^{14} + 8217916337135 p^{8} T^{16} - 13437805244 p^{12} T^{18} + 10961922 p^{16} T^{20} - 4972 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 160 T + 16590 T^{2} + 1165376 T^{3} + 65989583 T^{4} + 3021332256 T^{5} + 121125132964 T^{6} + 3021332256 p^{2} T^{7} + 65989583 p^{4} T^{8} + 1165376 p^{6} T^{9} + 16590 p^{8} T^{10} + 160 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 68 T + 5858 T^{2} - 324724 T^{3} + 19024079 T^{4} - 849006984 T^{5} + 41350350300 T^{6} - 849006984 p^{2} T^{7} + 19024079 p^{4} T^{8} - 324724 p^{6} T^{9} + 5858 p^{8} T^{10} - 68 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 - 13068 T^{2} + 75574882 T^{4} - 250048268572 T^{6} + 512715413915279 T^{8} - 701259311385534488 T^{10} + \)\(96\!\cdots\!20\)\( T^{12} - 701259311385534488 p^{4} T^{14} + 512715413915279 p^{8} T^{16} - 250048268572 p^{12} T^{18} + 75574882 p^{16} T^{20} - 13068 p^{20} T^{22} + p^{24} T^{24} \) | |
| 47 | \( 1 - 10924 T^{2} + 58431234 T^{4} - 214926051836 T^{6} + 655306845733487 T^{8} - 1793335598374155864 T^{10} + \)\(42\!\cdots\!84\)\( T^{12} - 1793335598374155864 p^{4} T^{14} + 655306845733487 p^{8} T^{16} - 214926051836 p^{12} T^{18} + 58431234 p^{16} T^{20} - 10924 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 - 88 T + 10190 T^{2} - 617912 T^{3} + 50451791 T^{4} - 2411920176 T^{5} + 159462242340 T^{6} - 2411920176 p^{2} T^{7} + 50451791 p^{4} T^{8} - 617912 p^{6} T^{9} + 10190 p^{8} T^{10} - 88 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 20636 T^{2} + 239872994 T^{4} - 1918291659628 T^{6} + 11645626899883535 T^{8} - 55752582720716753976 T^{10} + \)\(21\!\cdots\!76\)\( T^{12} - 55752582720716753976 p^{4} T^{14} + 11645626899883535 p^{8} T^{16} - 1918291659628 p^{12} T^{18} + 239872994 p^{16} T^{20} - 20636 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 - 20 T + 5426 T^{2} - 31204 T^{3} + 25902623 T^{4} - 243347112 T^{5} + 120789957756 T^{6} - 243347112 p^{2} T^{7} + 25902623 p^{4} T^{8} - 31204 p^{6} T^{9} + 5426 p^{8} T^{10} - 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 - 38924 T^{2} + 717401634 T^{4} - 8393249238940 T^{6} + 70361550899953487 T^{8} - \)\(45\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{12} - \)\(45\!\cdots\!68\)\( p^{4} T^{14} + 70361550899953487 p^{8} T^{16} - 8393249238940 p^{12} T^{18} + 717401634 p^{16} T^{20} - 38924 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( 1 - 24332 T^{2} + 305543810 T^{4} - 2552274795100 T^{6} + 15844798830822191 T^{8} - 81882082692758966040 T^{10} + \)\(40\!\cdots\!04\)\( T^{12} - 81882082692758966040 p^{4} T^{14} + 15844798830822191 p^{8} T^{16} - 2552274795100 p^{12} T^{18} + 305543810 p^{16} T^{20} - 24332 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 + 224 T + 40454 T^{2} + 5143648 T^{3} + 560729327 T^{4} + 50931113664 T^{5} + 4020188032404 T^{6} + 50931113664 p^{2} T^{7} + 560729327 p^{4} T^{8} + 5143648 p^{6} T^{9} + 40454 p^{8} T^{10} + 224 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 - 29036 T^{2} + 470607746 T^{4} - 5098003651516 T^{6} + 41939723612775791 T^{8} - \)\(28\!\cdots\!36\)\( T^{10} + \)\(18\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!36\)\( p^{4} T^{14} + 41939723612775791 p^{8} T^{16} - 5098003651516 p^{12} T^{18} + 470607746 p^{16} T^{20} - 29036 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 - 59852 T^{2} + 1680666146 T^{4} - 29638085732188 T^{6} + 371823834735989711 T^{8} - \)\(35\!\cdots\!64\)\( T^{10} + \)\(27\!\cdots\!12\)\( T^{12} - \)\(35\!\cdots\!64\)\( p^{4} T^{14} + 371823834735989711 p^{8} T^{16} - 29638085732188 p^{12} T^{18} + 1680666146 p^{16} T^{20} - 59852 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( ( 1 - 4 T + 26562 T^{2} - 843956 T^{3} + 354613391 T^{4} - 14829522696 T^{5} + 3312833571868 T^{6} - 14829522696 p^{2} T^{7} + 354613391 p^{4} T^{8} - 843956 p^{6} T^{9} + 26562 p^{8} T^{10} - 4 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 112 T + 34694 T^{2} + 2498096 T^{3} + 547109519 T^{4} + 33099463776 T^{5} + 6140319432468 T^{6} + 33099463776 p^{2} T^{7} + 547109519 p^{4} T^{8} + 2498096 p^{6} T^{9} + 34694 p^{8} T^{10} + 112 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−2.52525817980867554379913725326, −2.33583808538765407744985817459, −2.30118700029722775944102596669, −2.20346773090912233485478405211, −2.19614262641113716435180771420, −2.16227400470512963947303149559, −2.01705082680750097898195756376, −2.00061084691006413130572957361, −1.88513666582861606906471786605, −1.78437472774248045365821697020, −1.67377402850498233373253276306, −1.60882714198729570813656206725, −1.51616401114287098537196929382, −1.22107692779255092233648726574, −1.17338554307169298497040489873, −1.04420744916443635974343124701, −1.01734944831714622242821341590, −0.885987587019359331648226881050, −0.881526949983670672544354644899, −0.55819606189184190327117983457, −0.38348851695526253635733188876, −0.35746124998305942759572269644, −0.25448686168688357542028227468, −0.21093988896685576201269847742, −0.05242908575948620063649259921, 0.05242908575948620063649259921, 0.21093988896685576201269847742, 0.25448686168688357542028227468, 0.35746124998305942759572269644, 0.38348851695526253635733188876, 0.55819606189184190327117983457, 0.881526949983670672544354644899, 0.885987587019359331648226881050, 1.01734944831714622242821341590, 1.04420744916443635974343124701, 1.17338554307169298497040489873, 1.22107692779255092233648726574, 1.51616401114287098537196929382, 1.60882714198729570813656206725, 1.67377402850498233373253276306, 1.78437472774248045365821697020, 1.88513666582861606906471786605, 2.00061084691006413130572957361, 2.01705082680750097898195756376, 2.16227400470512963947303149559, 2.19614262641113716435180771420, 2.20346773090912233485478405211, 2.30118700029722775944102596669, 2.33583808538765407744985817459, 2.52525817980867554379913725326
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.