Dirichlet series
| L(s) = 1 | − 29·4-s + 44·7-s + 64·13-s + 381·16-s + 148·19-s − 1.27e3·28-s − 28·31-s + 648·37-s + 520·43-s − 992·49-s − 1.85e3·52-s + 52·61-s − 3.07e3·64-s + 2.72e3·67-s + 1.82e3·73-s − 4.29e3·76-s + 140·79-s + 2.81e3·91-s + 3.70e3·97-s + 6.10e3·103-s + 2.26e3·109-s + 1.67e4·112-s − 8.77e3·121-s + 812·124-s + 127-s + 131-s + 6.51e3·133-s + ⋯ |
| L(s) = 1 | − 3.62·4-s + 2.37·7-s + 1.36·13-s + 5.95·16-s + 1.78·19-s − 8.61·28-s − 0.162·31-s + 2.87·37-s + 1.84·43-s − 2.89·49-s − 4.94·52-s + 0.109·61-s − 6.00·64-s + 4.95·67-s + 2.93·73-s − 6.47·76-s + 0.199·79-s + 3.24·91-s + 3.88·97-s + 5.84·103-s + 1.99·109-s + 14.1·112-s − 6.59·121-s + 0.588·124-s + 0.000698·127-s + 0.000666·131-s + 4.24·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{48} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.46239\times 10^{24}\) |
| Root analytic conductor: | \(10.9306\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{48} \cdot 5^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(154.0869262\) |
| \(L(\frac12)\) | \(\approx\) | \(154.0869262\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + 29 T^{2} + 115 p^{2} T^{4} + 5365 T^{6} + 56089 T^{8} + 68623 p^{3} T^{10} + 298097 p^{4} T^{12} + 68623 p^{9} T^{14} + 56089 p^{12} T^{16} + 5365 p^{18} T^{18} + 115 p^{26} T^{20} + 29 p^{30} T^{22} + p^{36} T^{24} \) |
| 7 | \( ( 1 - 22 T + 1222 T^{2} - 3830 p T^{3} + 738277 T^{4} - 16429468 T^{5} + 304345091 T^{6} - 16429468 p^{3} T^{7} + 738277 p^{6} T^{8} - 3830 p^{10} T^{9} + 1222 p^{12} T^{10} - 22 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 11 | \( 1 + 8774 T^{2} + 38406937 T^{4} + 110794480912 T^{6} + 237518455614196 T^{8} + 407397597405316406 T^{10} + \)\(58\!\cdots\!20\)\( T^{12} + 407397597405316406 p^{6} T^{14} + 237518455614196 p^{12} T^{16} + 110794480912 p^{18} T^{18} + 38406937 p^{24} T^{20} + 8774 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 - 32 T + 5615 T^{2} - 22440 T^{3} + 8656798 T^{4} + 526171562 T^{5} + 4711694950 T^{6} + 526171562 p^{3} T^{7} + 8656798 p^{6} T^{8} - 22440 p^{9} T^{9} + 5615 p^{12} T^{10} - 32 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 17 | \( 1 + 25292 T^{2} + 350408026 T^{4} + 3357407842894 T^{6} + 24871140257230315 T^{8} + \)\(15\!\cdots\!62\)\( T^{10} + \)\(79\!\cdots\!29\)\( T^{12} + \)\(15\!\cdots\!62\)\( p^{6} T^{14} + 24871140257230315 p^{12} T^{16} + 3357407842894 p^{18} T^{18} + 350408026 p^{24} T^{20} + 25292 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 - 74 T + 26900 T^{2} - 1726326 T^{3} + 363105127 T^{4} - 19231149652 T^{5} + 3050951929144 T^{6} - 19231149652 p^{3} T^{7} + 363105127 p^{6} T^{8} - 1726326 p^{9} T^{9} + 26900 p^{12} T^{10} - 74 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 87606 T^{2} + 3901212549 T^{4} + 115936155044240 T^{6} + 110899212712323840 p T^{8} + \)\(43\!\cdots\!06\)\( T^{10} + \)\(59\!\cdots\!16\)\( T^{12} + \)\(43\!\cdots\!06\)\( p^{6} T^{14} + 110899212712323840 p^{13} T^{16} + 115936155044240 p^{18} T^{18} + 3901212549 p^{24} T^{20} + 87606 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( 1 + 90164 T^{2} + 5538725158 T^{4} + 241687583125540 T^{6} + 8907411854557821103 T^{8} + \)\(27\!\cdots\!00\)\( T^{10} + \)\(72\!\cdots\!92\)\( T^{12} + \)\(27\!\cdots\!00\)\( p^{6} T^{14} + 8907411854557821103 p^{12} T^{16} + 241687583125540 p^{18} T^{18} + 5538725158 p^{24} T^{20} + 90164 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( ( 1 + 14 T + 2782 T^{2} - 1444412 T^{3} + 1616217718 T^{4} - 35329049842 T^{5} - 1030392946702 T^{6} - 35329049842 p^{3} T^{7} + 1616217718 p^{6} T^{8} - 1444412 p^{9} T^{9} + 2782 p^{12} T^{10} + 14 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 324 T + 257343 T^{2} - 61005292 T^{3} + 27549071142 T^{4} - 5150159444442 T^{5} + 1734672041555574 T^{6} - 5150159444442 p^{3} T^{7} + 27549071142 p^{6} T^{8} - 61005292 p^{9} T^{9} + 257343 p^{12} T^{10} - 324 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( 1 + 516980 T^{2} + 116162332402 T^{4} + 14455440997222174 T^{6} + \)\(10\!\cdots\!23\)\( T^{8} + \)\(40\!\cdots\!42\)\( T^{10} + \)\(14\!\cdots\!29\)\( T^{12} + \)\(40\!\cdots\!42\)\( p^{6} T^{14} + \)\(10\!\cdots\!23\)\( p^{12} T^{16} + 14455440997222174 p^{18} T^{18} + 116162332402 p^{24} T^{20} + 516980 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 - 260 T + 419093 T^{2} - 95452830 T^{3} + 77517369766 T^{4} - 14557938541390 T^{5} + 8030790795328396 T^{6} - 14557938541390 p^{3} T^{7} + 77517369766 p^{6} T^{8} - 95452830 p^{9} T^{9} + 419093 p^{12} T^{10} - 260 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( 1 + 250826 T^{2} + 64468294693 T^{4} + 11227249671199678 T^{6} + \)\(16\!\cdots\!58\)\( T^{8} + \)\(22\!\cdots\!22\)\( T^{10} + \)\(23\!\cdots\!57\)\( T^{12} + \)\(22\!\cdots\!22\)\( p^{6} T^{14} + \)\(16\!\cdots\!58\)\( p^{12} T^{16} + 11227249671199678 p^{18} T^{18} + 64468294693 p^{24} T^{20} + 250826 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 + 1089914 T^{2} + 570627348181 T^{4} + 195167846011351672 T^{6} + \)\(49\!\cdots\!20\)\( T^{8} + \)\(10\!\cdots\!38\)\( T^{10} + \)\(16\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!38\)\( p^{6} T^{14} + \)\(49\!\cdots\!20\)\( p^{12} T^{16} + 195167846011351672 p^{18} T^{18} + 570627348181 p^{24} T^{20} + 1089914 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( 1 + 1410782 T^{2} + 945582368857 T^{4} + 409327522630461640 T^{6} + \)\(13\!\cdots\!44\)\( T^{8} + \)\(34\!\cdots\!10\)\( T^{10} + \)\(21\!\cdots\!36\)\( p^{2} T^{12} + \)\(34\!\cdots\!10\)\( p^{6} T^{14} + \)\(13\!\cdots\!44\)\( p^{12} T^{16} + 409327522630461640 p^{18} T^{18} + 945582368857 p^{24} T^{20} + 1410782 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 26 T + 890093 T^{2} - 24067248 T^{3} + 411098122156 T^{4} - 9701168980990 T^{5} + 115854509855423452 T^{6} - 9701168980990 p^{3} T^{7} + 411098122156 p^{6} T^{8} - 24067248 p^{9} T^{9} + 890093 p^{12} T^{10} - 26 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 1360 T + 1879924 T^{2} - 1713723290 T^{3} + 1397089207930 T^{4} - 951263512377400 T^{5} + 557408863531277012 T^{6} - 951263512377400 p^{3} T^{7} + 1397089207930 p^{6} T^{8} - 1713723290 p^{9} T^{9} + 1879924 p^{12} T^{10} - 1360 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( 1 + 1453688 T^{2} + 1137013985704 T^{4} + 692748586851362596 T^{6} + \)\(35\!\cdots\!64\)\( T^{8} + \)\(15\!\cdots\!88\)\( T^{10} + \)\(58\!\cdots\!82\)\( T^{12} + \)\(15\!\cdots\!88\)\( p^{6} T^{14} + \)\(35\!\cdots\!64\)\( p^{12} T^{16} + 692748586851362596 p^{18} T^{18} + 1137013985704 p^{24} T^{20} + 1453688 p^{30} T^{22} + p^{36} T^{24} \) | |
| 73 | \( ( 1 - 914 T + 2355269 T^{2} - 1654551222 T^{3} + 2285588896714 T^{4} - 1242988808966746 T^{5} + 1183779039908705629 T^{6} - 1242988808966746 p^{3} T^{7} + 2285588896714 p^{6} T^{8} - 1654551222 p^{9} T^{9} + 2355269 p^{12} T^{10} - 914 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 70 T + 336763 T^{2} + 344031346 T^{3} + 444692760427 T^{4} + 119555418660392 T^{5} + 200417710830066866 T^{6} + 119555418660392 p^{3} T^{7} + 444692760427 p^{6} T^{8} + 344031346 p^{9} T^{9} + 336763 p^{12} T^{10} - 70 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 + 4590744 T^{2} + 10478386779708 T^{4} + 15647429936711369156 T^{6} + \)\(16\!\cdots\!48\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{10} + \)\(90\!\cdots\!14\)\( T^{12} + \)\(14\!\cdots\!84\)\( p^{6} T^{14} + \)\(16\!\cdots\!48\)\( p^{12} T^{16} + 15647429936711369156 p^{18} T^{18} + 10478386779708 p^{24} T^{20} + 4590744 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( 1 + 3968934 T^{2} + 7587240468087 T^{4} + 9234899137375336814 T^{6} + \)\(81\!\cdots\!95\)\( T^{8} + \)\(58\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!98\)\( T^{12} + \)\(58\!\cdots\!92\)\( p^{6} T^{14} + \)\(81\!\cdots\!95\)\( p^{12} T^{16} + 9234899137375336814 p^{18} T^{18} + 7587240468087 p^{24} T^{20} + 3968934 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 - 1854 T + 3388047 T^{2} - 3940905670 T^{3} + 5514629779707 T^{4} - 5797874063043516 T^{5} + 6474785443353956586 T^{6} - 5797874063043516 p^{3} T^{7} + 5514629779707 p^{6} T^{8} - 3940905670 p^{9} T^{9} + 3388047 p^{12} T^{10} - 1854 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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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
−2.42975277136910023609908894621, −2.37805181028353986740827228887, −2.25224930780978635740653146783, −2.12140930094596940227598773470, −2.03198566332044060326482625065, −1.87858333905499745709819760238, −1.80741068291966488872553640051, −1.75825044682776175618399677021, −1.59983382569197914183835192816, −1.51553439847587443304336800375, −1.46542337440656548248312466828, −1.45114121393679965693468104369, −1.43516937799867730048343690820, −1.17280667563022965549792676007, −0.997335495804869553587029896955, −0.860174267645073994213952443998, −0.822614735757052768207131249984, −0.67330159075845042784680551315, −0.65979388748966679922382696029, −0.64650335319610496798033263253, −0.48276500961835311240866537482, −0.46871117112542314159900510015, −0.44086396117224349610098335373, −0.31783039339734035330177651950, −0.23071664790531944363066636544, 0.23071664790531944363066636544, 0.31783039339734035330177651950, 0.44086396117224349610098335373, 0.46871117112542314159900510015, 0.48276500961835311240866537482, 0.64650335319610496798033263253, 0.65979388748966679922382696029, 0.67330159075845042784680551315, 0.822614735757052768207131249984, 0.860174267645073994213952443998, 0.997335495804869553587029896955, 1.17280667563022965549792676007, 1.43516937799867730048343690820, 1.45114121393679965693468104369, 1.46542337440656548248312466828, 1.51553439847587443304336800375, 1.59983382569197914183835192816, 1.75825044682776175618399677021, 1.80741068291966488872553640051, 1.87858333905499745709819760238, 2.03198566332044060326482625065, 2.12140930094596940227598773470, 2.25224930780978635740653146783, 2.37805181028353986740827228887, 2.42975277136910023609908894621
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.