Dirichlet series
| L(s) = 1 | − 13·4-s − 57·9-s − 28·11-s + 135·16-s + 656·19-s − 670·29-s + 704·31-s + 741·36-s − 374·41-s + 364·44-s − 599·49-s − 596·59-s + 2.87e3·61-s − 46·64-s + 280·71-s − 8.52e3·76-s − 764·79-s + 999·81-s − 6.87e3·89-s + 1.59e3·99-s + 372·101-s + 6.13e3·109-s + 8.71e3·116-s + 2.55e3·121-s − 9.15e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.62·4-s − 2.11·9-s − 0.767·11-s + 2.10·16-s + 7.92·19-s − 4.29·29-s + 4.07·31-s + 3.43·36-s − 1.42·41-s + 1.24·44-s − 1.74·49-s − 1.31·59-s + 6.04·61-s − 0.0898·64-s + 0.468·71-s − 12.8·76-s − 1.08·79-s + 1.37·81-s − 8.18·89-s + 1.62·99-s + 0.366·101-s + 5.38·109-s + 6.97·116-s + 1.91·121-s − 6.62·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.99628\times 10^{13}\) |
| Root analytic conductor: | \(3.64354\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(6.199989429\) |
| \(L(\frac12)\) | \(\approx\) | \(6.199989429\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 19 p T^{2} + 250 p^{2} T^{4} + 319 p^{5} T^{6} + 250 p^{8} T^{8} + 19 p^{13} T^{10} + p^{18} T^{12} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + 13 T^{2} + 17 p T^{4} - 1267 T^{6} - 5029 p T^{8} + 32593 T^{10} + 948785 T^{12} + 32593 p^{6} T^{14} - 5029 p^{13} T^{16} - 1267 p^{18} T^{18} + 17 p^{25} T^{20} + 13 p^{30} T^{22} + p^{36} T^{24} \) |
| 7 | \( 1 + 599 T^{2} + 98807 T^{4} - 23535648 T^{6} - 16683134243 T^{8} - 14719255457 p^{3} T^{10} - 1627528496195738 T^{12} - 14719255457 p^{9} T^{14} - 16683134243 p^{12} T^{16} - 23535648 p^{18} T^{18} + 98807 p^{24} T^{20} + 599 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( ( 1 + 14 T - 981 T^{2} - 145154 T^{3} - 1312682 T^{4} + 6914350 p T^{5} + 9056022383 T^{6} + 6914350 p^{4} T^{7} - 1312682 p^{6} T^{8} - 145154 p^{9} T^{9} - 981 p^{12} T^{10} + 14 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 13 | \( 1 + 6678 T^{2} + 17309277 T^{4} + 40300870738 T^{6} + 138978347444874 T^{8} + 332728419646284318 T^{10} + \)\(62\!\cdots\!29\)\( T^{12} + 332728419646284318 p^{6} T^{14} + 138978347444874 p^{12} T^{16} + 40300870738 p^{18} T^{18} + 17309277 p^{24} T^{20} + 6678 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( ( 1 - 19762 T^{2} + 198791535 T^{4} - 1212050905772 T^{6} + 198791535 p^{6} T^{8} - 19762 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 164 T + 27869 T^{2} - 2307068 T^{3} + 27869 p^{3} T^{4} - 164 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 23 | \( 1 + 34095 T^{2} + 791165199 T^{4} + 7581006323032 T^{6} - 3677153771379843 T^{8} - \)\(20\!\cdots\!71\)\( T^{10} - \)\(32\!\cdots\!90\)\( T^{12} - \)\(20\!\cdots\!71\)\( p^{6} T^{14} - 3677153771379843 p^{12} T^{16} + 7581006323032 p^{18} T^{18} + 791165199 p^{24} T^{20} + 34095 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 + 335 T + 11727 T^{2} + 1199704 T^{3} + 2210558677 T^{4} + 188505793169 T^{5} - 20536380542074 T^{6} + 188505793169 p^{3} T^{7} + 2210558677 p^{6} T^{8} + 1199704 p^{9} T^{9} + 11727 p^{12} T^{10} + 335 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 352 T + 48463 T^{2} - 4331384 T^{3} - 467515598 T^{4} + 431517878828 T^{5} - 103162000220977 T^{6} + 431517878828 p^{3} T^{7} - 467515598 p^{6} T^{8} - 4331384 p^{9} T^{9} + 48463 p^{12} T^{10} - 352 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 191730 T^{2} + 19047314679 T^{4} - 1194553172305132 T^{6} + 19047314679 p^{6} T^{8} - 191730 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 187 T - 127197 T^{2} - 6771316 T^{3} + 11652659197 T^{4} - 283910950583 T^{5} - 979779052450642 T^{6} - 283910950583 p^{3} T^{7} + 11652659197 p^{6} T^{8} - 6771316 p^{9} T^{9} - 127197 p^{12} T^{10} + 187 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 43 | \( 1 + 134550 T^{2} + 13510973709 T^{4} + 1026189095282482 T^{6} + 454070353757653854 p T^{8} - \)\(33\!\cdots\!06\)\( T^{10} - \)\(30\!\cdots\!15\)\( T^{12} - \)\(33\!\cdots\!06\)\( p^{6} T^{14} + 454070353757653854 p^{13} T^{16} + 1026189095282482 p^{18} T^{18} + 13510973709 p^{24} T^{20} + 134550 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( 1 + 371275 T^{2} + 74900409715 T^{4} + 8773428375826940 T^{6} + \)\(57\!\cdots\!85\)\( T^{8} - \)\(12\!\cdots\!75\)\( T^{10} - \)\(29\!\cdots\!22\)\( T^{12} - \)\(12\!\cdots\!75\)\( p^{6} T^{14} + \)\(57\!\cdots\!85\)\( p^{12} T^{16} + 8773428375826940 p^{18} T^{18} + 74900409715 p^{24} T^{20} + 371275 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( ( 1 - 572962 T^{2} + 148278111975 T^{4} - 25340688789932924 T^{6} + 148278111975 p^{6} T^{8} - 572962 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 59 | \( ( 1 + 298 T - 172689 T^{2} + 87864938 T^{3} + 16154481778 T^{4} - 20738106147710 T^{5} - 438633693159913 T^{6} - 20738106147710 p^{3} T^{7} + 16154481778 p^{6} T^{8} + 87864938 p^{9} T^{9} - 172689 p^{12} T^{10} + 298 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 1439 T + 800471 T^{2} - 410160120 T^{3} + 305155978465 T^{4} - 144780863269729 T^{5} + 52951494992155726 T^{6} - 144780863269729 p^{3} T^{7} + 305155978465 p^{6} T^{8} - 410160120 p^{9} T^{9} + 800471 p^{12} T^{10} - 1439 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 + 573131 T^{2} + 29023042523 T^{4} - 2906842697978268 T^{6} + \)\(12\!\cdots\!09\)\( T^{8} - \)\(22\!\cdots\!99\)\( T^{10} - \)\(17\!\cdots\!10\)\( T^{12} - \)\(22\!\cdots\!99\)\( p^{6} T^{14} + \)\(12\!\cdots\!09\)\( p^{12} T^{16} - 2906842697978268 p^{18} T^{18} + 29023042523 p^{24} T^{20} + 573131 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( ( 1 - 70 T + 388273 T^{2} + 173667512 T^{3} + 388273 p^{3} T^{4} - 70 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 73 | \( ( 1 - 1452822 T^{2} + 1137063900063 T^{4} - 544488665125947316 T^{6} + 1137063900063 p^{6} T^{8} - 1452822 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 382 T - 945269 T^{2} - 59882698 T^{3} + 561648238738 T^{4} - 58077227393690 T^{5} - 326016591815072533 T^{6} - 58077227393690 p^{3} T^{7} + 561648238738 p^{6} T^{8} - 59882698 p^{9} T^{9} - 945269 p^{12} T^{10} + 382 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 + 418215 T^{2} - 775623191505 T^{4} - 150031963858198280 T^{6} + \)\(47\!\cdots\!05\)\( T^{8} + \)\(45\!\cdots\!65\)\( T^{10} - \)\(16\!\cdots\!82\)\( T^{12} + \)\(45\!\cdots\!65\)\( p^{6} T^{14} + \)\(47\!\cdots\!05\)\( p^{12} T^{16} - 150031963858198280 p^{18} T^{18} - 775623191505 p^{24} T^{20} + 418215 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( ( 1 + 1719 T + 2353398 T^{2} + 2298177027 T^{3} + 2353398 p^{3} T^{4} + 1719 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 97 | \( 1 + 5261178 T^{2} + 15955691977749 T^{4} + 33371126998180290382 T^{6} + \)\(53\!\cdots\!38\)\( T^{8} + \)\(66\!\cdots\!98\)\( T^{10} + \)\(67\!\cdots\!85\)\( T^{12} + \)\(66\!\cdots\!98\)\( p^{6} T^{14} + \)\(53\!\cdots\!38\)\( p^{12} T^{16} + 33371126998180290382 p^{18} T^{18} + 15955691977749 p^{24} T^{20} + 5261178 p^{30} T^{22} + p^{36} T^{24} \) | |
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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
−3.69721040598352272707578091374, −3.37205702275415889461564741219, −3.36045552467544379144734660864, −3.28812653351995441040447195374, −3.28775264104954091612799476720, −3.15971426531087709991911783470, −3.12932373122537849466292956630, −2.76601589535534873282365207288, −2.75444455338161256012979145838, −2.74962982987454283200524001505, −2.69794818999565558944134036667, −2.35444409267646759148766014905, −2.14226380213553347537707675049, −1.96221853897932516275872343155, −1.77789298359951485185831749657, −1.69641686854695994779360490484, −1.58271606894664532010706372383, −1.21423754510935957614729981842, −1.02843462551580343210086742203, −0.991676195477600213502830801949, −0.916483328219729323218030929727, −0.69054609298924393223449848816, −0.55100565006499660486143982023, −0.25724540645996849083565586677, −0.24864725981101670642112071851, 0.24864725981101670642112071851, 0.25724540645996849083565586677, 0.55100565006499660486143982023, 0.69054609298924393223449848816, 0.916483328219729323218030929727, 0.991676195477600213502830801949, 1.02843462551580343210086742203, 1.21423754510935957614729981842, 1.58271606894664532010706372383, 1.69641686854695994779360490484, 1.77789298359951485185831749657, 1.96221853897932516275872343155, 2.14226380213553347537707675049, 2.35444409267646759148766014905, 2.69794818999565558944134036667, 2.74962982987454283200524001505, 2.75444455338161256012979145838, 2.76601589535534873282365207288, 3.12932373122537849466292956630, 3.15971426531087709991911783470, 3.28775264104954091612799476720, 3.28812653351995441040447195374, 3.36045552467544379144734660864, 3.37205702275415889461564741219, 3.69721040598352272707578091374
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.