Dirichlet series
| L(s) = 1 | − 2-s + 3·4-s + 10·7-s − 11·8-s − 10·14-s + 23·16-s − 84·19-s + 642·25-s + 30·28-s − 200·29-s − 384·31-s − 7·32-s − 244·37-s + 84·38-s − 280·47-s − 162·49-s − 642·50-s + 16·53-s − 110·56-s + 200·58-s − 1.16e3·59-s + 384·62-s + 511·64-s + 244·74-s − 252·76-s + 968·83-s + 280·94-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 3/8·4-s + 0.539·7-s − 0.486·8-s − 0.190·14-s + 0.359·16-s − 1.01·19-s + 5.13·25-s + 0.202·28-s − 1.28·29-s − 2.22·31-s − 0.0386·32-s − 1.08·37-s + 0.358·38-s − 0.868·47-s − 0.472·49-s − 1.81·50-s + 0.0414·53-s − 0.262·56-s + 0.452·58-s − 2.57·59-s + 0.786·62-s + 0.998·64-s + 0.383·74-s − 0.380·76-s + 1.28·83-s + 0.307·94-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.16734\times 10^{14}\) |
| Root analytic conductor: | \(3.85596\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(2.353078811\) |
| \(L(\frac12)\) | \(\approx\) | \(2.353078811\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T - p T^{2} + 3 p T^{3} - 7 p^{3} T^{5} - 7 p^{6} T^{6} - 7 p^{6} T^{7} + 3 p^{10} T^{9} - p^{13} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 10 T + 262 T^{2} + 842 T^{3} + 12281 p T^{4} - 27388 p^{2} T^{5} + 282796 p^{3} T^{6} - 27388 p^{5} T^{7} + 12281 p^{7} T^{8} + 842 p^{9} T^{9} + 262 p^{12} T^{10} - 10 p^{15} T^{11} + p^{18} T^{12} \) | |
| good | 5 | \( 1 - 642 T^{2} + 208394 T^{4} - 9322026 p T^{6} + 8351455471 T^{8} - 1283212969212 T^{10} + 171916133103148 T^{12} - 1283212969212 p^{6} T^{14} + 8351455471 p^{12} T^{16} - 9322026 p^{19} T^{18} + 208394 p^{24} T^{20} - 642 p^{30} T^{22} + p^{36} T^{24} \) |
| 11 | \( 1 - 5558 T^{2} + 17328562 T^{4} - 39697781446 T^{6} + 74680549582767 T^{8} - 10925809255157404 p T^{10} + \)\(16\!\cdots\!76\)\( T^{12} - 10925809255157404 p^{7} T^{14} + 74680549582767 p^{12} T^{16} - 39697781446 p^{18} T^{18} + 17328562 p^{24} T^{20} - 5558 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( 1 - 16388 T^{2} + 130168450 T^{4} - 667979540948 T^{6} + 2502014595031295 T^{8} - 7350443941021209352 T^{10} + \)\(17\!\cdots\!68\)\( T^{12} - 7350443941021209352 p^{6} T^{14} + 2502014595031295 p^{12} T^{16} - 667979540948 p^{18} T^{18} + 130168450 p^{24} T^{20} - 16388 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( 1 - 24130 T^{2} + 336665322 T^{4} - 3317046729666 T^{6} + 25449379046911359 T^{8} - \)\(16\!\cdots\!44\)\( T^{10} + \)\(85\!\cdots\!96\)\( T^{12} - \)\(16\!\cdots\!44\)\( p^{6} T^{14} + 25449379046911359 p^{12} T^{16} - 3317046729666 p^{18} T^{18} + 336665322 p^{24} T^{20} - 24130 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 + 42 T + 14070 T^{2} - 395490 T^{3} + 3286109 p T^{4} - 9398728548 T^{5} + 289465860276 T^{6} - 9398728548 p^{3} T^{7} + 3286109 p^{7} T^{8} - 395490 p^{9} T^{9} + 14070 p^{12} T^{10} + 42 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 - 92822 T^{2} + 4403882674 T^{4} - 138019190867542 T^{6} + 3157265325811091679 T^{8} - \)\(55\!\cdots\!92\)\( T^{10} + \)\(75\!\cdots\!32\)\( T^{12} - \)\(55\!\cdots\!92\)\( p^{6} T^{14} + 3157265325811091679 p^{12} T^{16} - 138019190867542 p^{18} T^{18} + 4403882674 p^{24} T^{20} - 92822 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 + 100 T + 47898 T^{2} + 9341604 T^{3} + 2007541655 T^{4} + 290966053384 T^{5} + 66257486741772 T^{6} + 290966053384 p^{3} T^{7} + 2007541655 p^{6} T^{8} + 9341604 p^{9} T^{9} + 47898 p^{12} T^{10} + 100 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 192 T + 79514 T^{2} + 6935616 T^{3} + 3231616847 T^{4} + 390779082624 T^{5} + 138834590310572 T^{6} + 390779082624 p^{3} T^{7} + 3231616847 p^{6} T^{8} + 6935616 p^{9} T^{9} + 79514 p^{12} T^{10} + 192 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 122 T + 230498 T^{2} + 34280946 T^{3} + 23640098487 T^{4} + 3642512537420 T^{5} + 1472029170650908 T^{6} + 3642512537420 p^{3} T^{7} + 23640098487 p^{6} T^{8} + 34280946 p^{9} T^{9} + 230498 p^{12} T^{10} + 122 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( 1 - 453906 T^{2} + 91605366794 T^{4} - 10752866720419890 T^{6} + \)\(81\!\cdots\!67\)\( T^{8} - \)\(44\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(44\!\cdots\!16\)\( p^{6} T^{14} + \)\(81\!\cdots\!67\)\( p^{12} T^{16} - 10752866720419890 p^{18} T^{18} + 91605366794 p^{24} T^{20} - 453906 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( 1 - 625992 T^{2} + 191021018714 T^{4} - 37826754729984360 T^{6} + \)\(54\!\cdots\!51\)\( T^{8} - \)\(60\!\cdots\!88\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(60\!\cdots\!88\)\( p^{6} T^{14} + \)\(54\!\cdots\!51\)\( p^{12} T^{16} - 37826754729984360 p^{18} T^{18} + 191021018714 p^{24} T^{20} - 625992 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( ( 1 + 140 T + 320330 T^{2} + 73212740 T^{3} + 60859580527 T^{4} + 11849141354936 T^{5} + 8088315502149964 T^{6} + 11849141354936 p^{3} T^{7} + 60859580527 p^{6} T^{8} + 73212740 p^{9} T^{9} + 320330 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 8 T + 620338 T^{2} - 43270728 T^{3} + 175052111079 T^{4} - 17854809544976 T^{5} + 31198306509520220 T^{6} - 17854809544976 p^{3} T^{7} + 175052111079 p^{6} T^{8} - 43270728 p^{9} T^{9} + 620338 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( ( 1 + 584 T + 660226 T^{2} + 288363320 T^{3} + 227118238535 T^{4} + 84061764799568 T^{5} + 53003863801684156 T^{6} + 84061764799568 p^{3} T^{7} + 227118238535 p^{6} T^{8} + 288363320 p^{9} T^{9} + 660226 p^{12} T^{10} + 584 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( 1 - 1132844 T^{2} + 598144576978 T^{4} - 187592974802688284 T^{6} + \)\(36\!\cdots\!15\)\( T^{8} - \)\(44\!\cdots\!20\)\( T^{10} + \)\(52\!\cdots\!48\)\( T^{12} - \)\(44\!\cdots\!20\)\( p^{6} T^{14} + \)\(36\!\cdots\!15\)\( p^{12} T^{16} - 187592974802688284 p^{18} T^{18} + 598144576978 p^{24} T^{20} - 1132844 p^{30} T^{22} + p^{36} T^{24} \) | |
| 67 | \( 1 - 2201456 T^{2} + 2446914850954 T^{4} - 1810143772474049072 T^{6} + \)\(98\!\cdots\!19\)\( T^{8} - \)\(41\!\cdots\!88\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{12} - \)\(41\!\cdots\!88\)\( p^{6} T^{14} + \)\(98\!\cdots\!19\)\( p^{12} T^{16} - 1810143772474049072 p^{18} T^{18} + 2446914850954 p^{24} T^{20} - 2201456 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( 1 - 41114 p T^{2} + 4283951535986 T^{4} - 4104693796296127926 T^{6} + \)\(28\!\cdots\!15\)\( T^{8} - \)\(14\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!48\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{6} T^{14} + \)\(28\!\cdots\!15\)\( p^{12} T^{16} - 4104693796296127926 p^{18} T^{18} + 4283951535986 p^{24} T^{20} - 41114 p^{31} T^{22} + p^{36} T^{24} \) | |
| 73 | \( 1 - 1439236 T^{2} + 1102003928274 T^{4} - 451513704340917812 T^{6} + \)\(43\!\cdots\!39\)\( T^{8} + \)\(61\!\cdots\!52\)\( T^{10} - \)\(39\!\cdots\!08\)\( T^{12} + \)\(61\!\cdots\!52\)\( p^{6} T^{14} + \)\(43\!\cdots\!39\)\( p^{12} T^{16} - 451513704340917812 p^{18} T^{18} + 1102003928274 p^{24} T^{20} - 1439236 p^{30} T^{22} + p^{36} T^{24} \) | |
| 79 | \( 1 - 2332448 T^{2} + 3331150435450 T^{4} - 3366161588747882144 T^{6} + \)\(26\!\cdots\!99\)\( T^{8} - \)\(17\!\cdots\!80\)\( T^{10} + \)\(93\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!80\)\( p^{6} T^{14} + \)\(26\!\cdots\!99\)\( p^{12} T^{16} - 3366161588747882144 p^{18} T^{18} + 3331150435450 p^{24} T^{20} - 2332448 p^{30} T^{22} + p^{36} T^{24} \) | |
| 83 | \( ( 1 - 484 T + 1270706 T^{2} - 875417788 T^{3} + 1295203458391 T^{4} - 787465212856936 T^{5} + 845488477438432156 T^{6} - 787465212856936 p^{3} T^{7} + 1295203458391 p^{6} T^{8} - 875417788 p^{9} T^{9} + 1270706 p^{12} T^{10} - 484 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 - 2324754 T^{2} + 3623829356426 T^{4} - 4250784438141992562 T^{6} + \)\(40\!\cdots\!59\)\( T^{8} - \)\(33\!\cdots\!12\)\( T^{10} + \)\(25\!\cdots\!84\)\( T^{12} - \)\(33\!\cdots\!12\)\( p^{6} T^{14} + \)\(40\!\cdots\!59\)\( p^{12} T^{16} - 4250784438141992562 p^{18} T^{18} + 3623829356426 p^{24} T^{20} - 2324754 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( 1 - 5179748 T^{2} + 14000284408306 T^{4} - 26536073421090570836 T^{6} + \)\(39\!\cdots\!39\)\( T^{8} - \)\(47\!\cdots\!40\)\( T^{10} + \)\(47\!\cdots\!48\)\( T^{12} - \)\(47\!\cdots\!40\)\( p^{6} T^{14} + \)\(39\!\cdots\!39\)\( p^{12} T^{16} - 26536073421090570836 p^{18} T^{18} + 14000284408306 p^{24} T^{20} - 5179748 p^{30} T^{22} + p^{36} T^{24} \) | |
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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
−3.58415659686293822948163038815, −3.49089054673366862182928767667, −3.31258296781964221911770242330, −3.29955416054073761995044520666, −3.19331634675668284045774869045, −2.83647061761157538630094380264, −2.81131174362307648565906204323, −2.80989386872649746892693862084, −2.66920251440876114800176485326, −2.62145203413580239635247992802, −2.51071359639766773604571765444, −2.04640919232128877835486925071, −1.92107999675924466189613051776, −1.86576076402784062512433501433, −1.81216503138450298834472863597, −1.68883083664318660673702655035, −1.65257225838441165855640533742, −1.52775793385551043345620892241, −1.00944354663513529057551413655, −0.891186900787480453931902936353, −0.855297038793582759731001468808, −0.800830088635151396195879915041, −0.56566836568425442112774076990, −0.17013813352285516130640906202, −0.15650476153987937019464980847, 0.15650476153987937019464980847, 0.17013813352285516130640906202, 0.56566836568425442112774076990, 0.800830088635151396195879915041, 0.855297038793582759731001468808, 0.891186900787480453931902936353, 1.00944354663513529057551413655, 1.52775793385551043345620892241, 1.65257225838441165855640533742, 1.68883083664318660673702655035, 1.81216503138450298834472863597, 1.86576076402784062512433501433, 1.92107999675924466189613051776, 2.04640919232128877835486925071, 2.51071359639766773604571765444, 2.62145203413580239635247992802, 2.66920251440876114800176485326, 2.80989386872649746892693862084, 2.81131174362307648565906204323, 2.83647061761157538630094380264, 3.19331634675668284045774869045, 3.29955416054073761995044520666, 3.31258296781964221911770242330, 3.49089054673366862182928767667, 3.58415659686293822948163038815
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.