Dirichlet series
| L(s) = 1 | − 16·4-s − 18·7-s − 154·13-s + 137·16-s − 48·19-s + 704·25-s + 288·28-s − 360·37-s − 810·43-s − 1.40e3·49-s + 2.46e3·52-s + 1.29e3·61-s − 612·64-s + 2.65e3·67-s + 768·76-s − 6.38e3·79-s + 2.77e3·91-s + 1.68e3·97-s − 1.12e4·100-s − 5.48e3·103-s − 2.46e3·112-s − 4.89e3·121-s + 127-s + 131-s + 864·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.971·7-s − 3.28·13-s + 2.14·16-s − 0.579·19-s + 5.63·25-s + 1.94·28-s − 1.59·37-s − 2.87·43-s − 4.08·49-s + 6.57·52-s + 2.71·61-s − 1.19·64-s + 4.84·67-s + 1.15·76-s − 9.08·79-s + 3.19·91-s + 1.76·97-s − 11.2·100-s − 5.24·103-s − 2.08·112-s − 3.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.563·133-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17117\times 10^{10}\) |
| Root analytic conductor: | \(2.62739\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 13^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.07313564293\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07313564293\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( ( 1 + 77 T + 110 p T^{2} + 5 p^{2} T^{3} + 110 p^{4} T^{4} + 77 p^{6} T^{5} + p^{9} T^{6} )^{2} \) | |
| good | 2 | \( 1 + p^{4} T^{2} + 119 T^{4} + 81 p^{2} T^{6} - 4943 T^{8} - 43541 p T^{10} - 213287 p^{2} T^{12} - 43541 p^{7} T^{14} - 4943 p^{12} T^{16} + 81 p^{20} T^{18} + 119 p^{24} T^{20} + p^{34} T^{22} + p^{36} T^{24} \) |
| 5 | \( ( 1 - 352 T^{2} + 67004 T^{4} - 9024514 T^{6} + 67004 p^{6} T^{8} - 352 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 7 | \( ( 1 + 9 T + 822 T^{2} + 7155 T^{3} + 7782 p^{2} T^{4} + 4765329 T^{5} + 150790930 T^{6} + 4765329 p^{3} T^{7} + 7782 p^{8} T^{8} + 7155 p^{9} T^{9} + 822 p^{12} T^{10} + 9 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 11 | \( 1 + 4894 T^{2} + 11449613 T^{4} + 20363323050 T^{6} + 34257579965962 T^{8} + 53736586813823798 T^{10} + 75632646342984402373 T^{12} + 53736586813823798 p^{6} T^{14} + 34257579965962 p^{12} T^{16} + 20363323050 p^{18} T^{18} + 11449613 p^{24} T^{20} + 4894 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( 1 - 17628 T^{2} + 148790580 T^{4} - 901368482188 T^{6} + 5101534626884604 T^{8} - 1802618725559253300 p T^{10} + \)\(16\!\cdots\!34\)\( T^{12} - 1802618725559253300 p^{7} T^{14} + 5101534626884604 p^{12} T^{16} - 901368482188 p^{18} T^{18} + 148790580 p^{24} T^{20} - 17628 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 + 24 T + 15837 T^{2} + 375480 T^{3} + 139880010 T^{4} + 3609456456 T^{5} + 1055020725565 T^{6} + 3609456456 p^{3} T^{7} + 139880010 p^{6} T^{8} + 375480 p^{9} T^{9} + 15837 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 - 12750 T^{2} - 179763339 T^{4} + 3502860139958 T^{6} + 16355064649567050 T^{8} - \)\(33\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!85\)\( T^{12} - \)\(33\!\cdots\!06\)\( p^{6} T^{14} + 16355064649567050 p^{12} T^{16} + 3502860139958 p^{18} T^{18} - 179763339 p^{24} T^{20} - 12750 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( 1 - 31380 T^{2} + 379534668 T^{4} - 10580424325276 T^{6} - 125688456487998108 T^{8} + \)\(89\!\cdots\!96\)\( T^{10} - \)\(12\!\cdots\!18\)\( T^{12} + \)\(89\!\cdots\!96\)\( p^{6} T^{14} - 125688456487998108 p^{12} T^{16} - 10580424325276 p^{18} T^{18} + 379534668 p^{24} T^{20} - 31380 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( ( 1 - 80727 T^{2} + 3957011115 T^{4} - 145324204911866 T^{6} + 3957011115 p^{6} T^{8} - 80727 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 180 T + 101892 T^{2} + 16396560 T^{3} + 5009846748 T^{4} + 1551571714332 T^{5} + 295519618730950 T^{6} + 1551571714332 p^{3} T^{7} + 5009846748 p^{6} T^{8} + 16396560 p^{9} T^{9} + 101892 p^{12} T^{10} + 180 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( 1 + 92032 T^{2} + 6340188836 T^{4} + 523007314414956 T^{6} + 9466867452432585076 T^{8} - \)\(13\!\cdots\!80\)\( T^{10} + \)\(28\!\cdots\!94\)\( T^{12} - \)\(13\!\cdots\!80\)\( p^{6} T^{14} + 9466867452432585076 p^{12} T^{16} + 523007314414956 p^{18} T^{18} + 6340188836 p^{24} T^{20} + 92032 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 + 405 T - 54090 T^{2} - 17527981 T^{3} + 8794911330 T^{4} + 58723713765 T^{5} - 1012328496429330 T^{6} + 58723713765 p^{3} T^{7} + 8794911330 p^{6} T^{8} - 17527981 p^{9} T^{9} - 54090 p^{12} T^{10} + 405 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( ( 1 - 463894 T^{2} + 102299363711 T^{4} - 13443518873535124 T^{6} + 102299363711 p^{6} T^{8} - 463894 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 123852 T^{2} + 20708854044 T^{4} + 6694409920272298 T^{6} + 20708854044 p^{6} T^{8} + 123852 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 + 1063186 T^{2} + 634098360125 T^{4} + 264679415553463110 T^{6} + \)\(85\!\cdots\!02\)\( T^{8} + \)\(22\!\cdots\!82\)\( T^{10} + \)\(50\!\cdots\!89\)\( T^{12} + \)\(22\!\cdots\!82\)\( p^{6} T^{14} + \)\(85\!\cdots\!02\)\( p^{12} T^{16} + 264679415553463110 p^{18} T^{18} + 634098360125 p^{24} T^{20} + 1063186 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 647 T - 76501 T^{2} + 79302564 T^{3} + 3470218741 T^{4} + 13154199398603 T^{5} - 13891561279304978 T^{6} + 13154199398603 p^{3} T^{7} + 3470218741 p^{6} T^{8} + 79302564 p^{9} T^{9} - 76501 p^{12} T^{10} - 647 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 1329 T + 1453206 T^{2} - 1148866011 T^{3} + 775255592850 T^{4} - 441705527825817 T^{5} + 257933371845374134 T^{6} - 441705527825817 p^{3} T^{7} + 775255592850 p^{6} T^{8} - 1148866011 p^{9} T^{9} + 1453206 p^{12} T^{10} - 1329 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( 1 + 15338 p T^{2} + 727271840981 T^{4} + 218382140099053458 T^{6} - \)\(14\!\cdots\!58\)\( T^{8} - \)\(45\!\cdots\!46\)\( T^{10} - \)\(22\!\cdots\!47\)\( T^{12} - \)\(45\!\cdots\!46\)\( p^{6} T^{14} - \)\(14\!\cdots\!58\)\( p^{12} T^{16} + 218382140099053458 p^{18} T^{18} + 727271840981 p^{24} T^{20} + 15338 p^{31} T^{22} + p^{36} T^{24} \) | |
| 73 | \( ( 1 - 2222037 T^{2} + 2098500922278 T^{4} - 1077893830511252705 T^{6} + 2098500922278 p^{6} T^{8} - 2222037 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 1595 T + 1551833 T^{2} + 1057580906 T^{3} + 1551833 p^{3} T^{4} + 1595 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 83 | \( ( 1 - 1171294 T^{2} + 910614134951 T^{4} - 510677804747973604 T^{6} + 910614134951 p^{6} T^{8} - 1171294 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 + 1411870 T^{2} + 932091177173 T^{4} + 55386162076548474 T^{6} - \)\(41\!\cdots\!78\)\( T^{8} - \)\(37\!\cdots\!94\)\( T^{10} - \)\(28\!\cdots\!43\)\( T^{12} - \)\(37\!\cdots\!94\)\( p^{6} T^{14} - \)\(41\!\cdots\!78\)\( p^{12} T^{16} + 55386162076548474 p^{18} T^{18} + 932091177173 p^{24} T^{20} + 1411870 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 - 843 T + 1725834 T^{2} - 1255185693 T^{3} + 1302244790208 T^{4} - 1651486867608783 T^{5} + 1138534054363123432 T^{6} - 1651486867608783 p^{3} T^{7} + 1302244790208 p^{6} T^{8} - 1255185693 p^{9} T^{9} + 1725834 p^{12} T^{10} - 843 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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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
−4.36769726646268844758593472444, −4.23782240738996043264900071987, −4.04213175947267956314969074388, −3.94551484475245193424086065461, −3.73970176753933639986015866536, −3.67368635731399929756088431338, −3.62418155634943380570763714675, −3.21926816042431421160588305666, −3.21733344005454700841603650009, −2.98892733072705219761173581880, −2.92575761076214194650279068561, −2.81247094028428406186459485802, −2.72426351842768423030022232885, −2.56907585091108900035883279565, −2.36884525268762563040080019890, −2.24550726583497969713645781002, −1.85513449892764155251562460182, −1.61218551927259065673444463452, −1.52572572456582849386574327776, −1.20924395657811730603387208675, −1.05967732123881563106536281267, −1.00089987257040828842941520445, −0.37592473692954580509090685607, −0.16552113298127580186628576779, −0.11358599699806861874353710714, 0.11358599699806861874353710714, 0.16552113298127580186628576779, 0.37592473692954580509090685607, 1.00089987257040828842941520445, 1.05967732123881563106536281267, 1.20924395657811730603387208675, 1.52572572456582849386574327776, 1.61218551927259065673444463452, 1.85513449892764155251562460182, 2.24550726583497969713645781002, 2.36884525268762563040080019890, 2.56907585091108900035883279565, 2.72426351842768423030022232885, 2.81247094028428406186459485802, 2.92575761076214194650279068561, 2.98892733072705219761173581880, 3.21733344005454700841603650009, 3.21926816042431421160588305666, 3.62418155634943380570763714675, 3.67368635731399929756088431338, 3.73970176753933639986015866536, 3.94551484475245193424086065461, 4.04213175947267956314969074388, 4.23782240738996043264900071987, 4.36769726646268844758593472444
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.